# Non zero sum games

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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 is an example 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 less than what they began with, such as in a game of poker played in a casino in which a cut is taken by the house.

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. Optimal strategies for two-player zero-sum games can often be found using 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 persons, 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.^{[1]}

### Psychology Edit

The most common or simple example 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 can pursue mutually destructive behavior as they choose their own ends.

## Economics and non-zero-sumEdit

Non-zero-sum situations are an important part of economic activity due to production, marginal utility and value . Most economic situations are non-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. One strategy for non-zero-sum games is tit for tat.

If a farmer succeeds in raising a bumper crop, he will benefit by being able to sell more food and make more money. The consumers he serves benefit as well, because there is more food to go around, so the price per unit of food will be lower. Other farmers who have not had such a good crop might suffer somewhat due to these lower prices, but this cost to other farmers may very well be less than the benefits enjoyed by everyone else, such that overall the bumper crop has created a net benefit. The same argument applies to other types of productive activity.

Trade is a non-zero-sum activity because all parties to a voluntary transaction believe that they will be better off after the trade than before, otherwise they would not participate. It is possible that they are mistaken in this belief, but experience suggests that people are more often than not able to judge correctly when a transaction would leave them better off, and thus persist in trading throughout their lives. It is not always the case that every participant will benefit equally. However, a trade is still a non-zero-sum situation whenever the result is a net gain, regardless of how evenly or unevenly that gain is distributed.

The non-zero-sum nature of economic transactions contrasts with the zero-sum nature in which they are reflected by the transfer of money. Some see this as a justification for advancing less traditional forms of money.

## Complexity and non-zero-sumEdit

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 one supporter of this view 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]

A criticism of this view points out that - in a true zero-sum system - win-win solutions are impossible for everyone to accomplish, balancing out in such a way to reduce a portion of the population to accepting lose-lose solutions. In other words, it is only possible to perceive a non-zero-sum system when not looking at the whole system (i.e. when observing the win-win portion of the population alone), but just one of its unbalanced parts. When the scope of observation is wide enough, each observed system will exhibit zero-sum characteristics.

## An exampleEdit

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 chooses in secret one of the two actions 1 or 2; the second player, 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.

## See alsoEdit

## ReferencesEdit

- ↑ This paragraph was translated from the French wikipedia article on this subject.

## Further readingEdit

### BooksEdit

- Knelman, J. (1996). How can I win if you don't lose? Games where the winner doesn't take all. Thousand Oaks, CA ; Maidenhead, BRK, England: Sage Publications, Inc; Open University Press.
- Nash, J. F., Jr. (2005). The Bargaining Problem. Northampton, MA: Edward Elgar Publishing.

### PapersEdit

- Ben-Ner, A., Putterman, L., Kong, F., & Magan, D. (2004). Reciprocity in a two-part dictator game: Journal of Economic Behavior & Organization Vol 53(3) Mar 2004, 333-352.
- Bierhoff, H. W. (1975). Commentary on Schwanenberg and Huth's "On the relevance of experimental non-zero-sum games." Zeitschrift fur Sozialpsychologie Vol 6(2) 1975, 172-175.
- Chen, K. (1975). Value of information in conflictual situations: Two simple examples: IEEE Transactions on Systems, Man, & Cybernetics Vol 5(4) Jul 1975, 466-470.
- Colman, A. M., & Stirk, J. A. (1998). Stackelberg reasoning in mixed-motive games: An experimental investigation: Journal of Economic Psychology Vol 19(2) Apr 1998, 279-293.
- Forges, F., & Minelli, E. (1997). A property of Nash equilibria in repeated games with incomplete information: Games and Economic Behavior Vol 18(2) Feb 1997, 159-175.
- Fox, J., & Guyer, M. (1973). Equivalence and stooge strategies in zero-sum games: Journal of Conflict Resolution Vol 17(3) Sep 1973, 513-533.
- Fox, S. (1976). Analysis of the individualistic, competitive and cooperative motives in non-zero-sum games: Psychological Reports Vol 39(1) Aug 1976, 55-61.
- Furntratt, E., & Graser, H. (1981). Free operant avoidance in a non-zero-sum communicative game: Psychologische Beitrage Vol 23(1) 1981, 136-145.
- Gossner, O., & Vieille, N. (2002). How to play with a biased coin? : Games and Economic Behavior Vol 41(2) Nov 2002, 206-226.
- Harris, C. (1998). On the Rate of Convergence of Continuous-Time Fictitious Play: Games and Economic Behavior Vol 22(2) Feb 1998, 238-259.
- Head, B. C. (1973). 3rd-person non-zero sum game theory: Dissertation Abstracts International Vol.
- Hessel, M. (1981). Bargaining costs and rational behavior: A simple model: Journal of Conflict Resolution Vol 25(3) Sep 1981, 535-558.
- Joosten, R., Peters, H., & Thuijsman, F. (1995). Unlearning by not doing: Repeated games and vanishing actions: Games and Economic Behavior Vol 9(1) Apr 1995, 1-7.
- Krivohlavy, J. (1970). Interpersonal relations in asymmetric situations: Ceskoslovenska Psychologie Vol 14(6) 1970, 544-567.
- Krivohlavy, J. (1976). The psychology of cooperation and creative interpersonal conflict resolution: Przeglad Psychologiczny Vol 19(4) 1976, 491-509.
- Lendenmann, K. W., & Rapoport, A. (1980). Decision pressures in 2x2 games: Behavioral Science Vol 25(2) Mar 1980, 107-119.
- Liebrand, W. B., Wilke, H. A., Vogel, R., & Wolters, F. J. (1986). Value orientation and conformity: A study using three types of social dilemma games: Journal of Conflict Resolution Vol 30(1) Mar 1986, 77-97.
- Mack, D. (1972). "Leader": An unbiased mixed-motive game? : Psychologische Beitrage Vol 14(2) 1972, 244-252.
- Markakis, E., & Saberi, A. (2005). On the core of the multicommodity flow game: Decision Support Systems Vol 39(1) Mar 2005, 3-10.
- Michener, H. A., Ginsberg, I. J., & Yuen, K. (1979). Effects of core properties in four-person games with side-payments: Behavioral Science Vol 24(4) Jul 1979, 263-280.
- Neyman, A., & Okada, D. (2000). Repeated Games with Bounded Entropy: Games and Economic Behavior Vol 30(2) Feb 2000, 228-247.
- Okamura, K., Kanaoka, T., Okada, T., & Tomita, S. (1984). Learning behavior of variable-structure stochastic automata in a three-person zero-sum game: IEEE Transactions on Systems, Man, & Cybernetics Vol SMC-14(6) Nov-Dec 1984, 924-932.
- Pate, J. L., Broughton, E. D., Hallman, L. K., & Letterman, N. L. (1974). Learning in two-person, zero-sum games: Psychological Reports Vol 34(2) Apr 1974, 503-510.
- Rabow, G. (1988). The cooperative edge: Psychology Today Vol 22(1) Jan 1988, 54-58.
- Rapoport, A., Kahan, J. P., & Stein, W. E. (1976). Decisions of timing in experimental probabilistic duels: Journal of Mathematical Psychology Vol 13(2) Apr 1976, 163-191.
- Reka, J., & Boglarka, S. (2005). Domain-specific Decisions in Non-zero Amount Games in Function of the Specialization: Erdelyi Pszichologiai Szemle Vol 6(4) Dec 2005, 361-384.
- Sato, K., Toda, M., & Yamagishi, T. (1985). A time series analysis of behavior in a social dilemma: Japanese Journal of Psychology Vol 56(5) Dec 1985, 277-283.
- Savchenko, T. N. (1987). Modelling of decision-making in a game of two persons with non-opposite interests: Psikologicheskii Zhurnal Vol 8(5) Sep-Oct 1987, 142-146.
- Schwanenberg, E., & Huth, W. (1974). The relevance of experimental non-zero-sum games: Zeitschrift fur Sozialpsychologie Vol 5(3) 1974, 167-183.
- Shim, J. P. (1988). The visual interactive computer-aided approach to teaching zero-sum/nonzero-sum games: Social Science Computer Review Vol 6(3) Fal 1988, 392-397.
- Shubik, M. (1978). Opinions on how one should play a three-person nonconstant sum game: Simulation & Games Vol 9(3) Sep 1978, 301-308.
- Shubik, M. (1986). Cooperative game solutions: Australian, Indian, and U.S. opinions: Journal of Conflict Resolution Vol 30(1) Mar 1986, 63-76.
- Simon, F. B. (1987). The social context of zero-sum-games and non-zero-sum-games: International Journal of Family Psychiatry Vol 8(3) 1987, 271-287.
- Simon, R. S. (2002). Separation of joint plan equilibrium payoffs from the min-max functions: Games and Economic Behavior Vol 41(1) Oct 2002, 79-102.
- Starr, J., & Yngvesson, B. (1975). Scarcity and disputing: Zeroing-in on compromise decisions: American Ethnologist Vol 2(3) Aug 1975, 553-566.
- Sundali, J. A., Rapoport, A., & Seale, D. A. (1995). Coordination in market entry games with symmetric players: Organizational Behavior and Human Decision Processes Vol 64(2) Nov 1995, 203-218.
- Teraoka, T. (1978). Effects of coalition in the symmetric three-person non-zero sum matrix game: Simulation analysis of the selection processes based upon the normative models of needs with reference to the martyr situation: Tohoku Psychologica Folia Vol 37(1-4) 1978, 131-144.
- Tversky, A. (1967). The Psychologist's Dilemma: PsycCRITIQUES Vol 12 (9), Sep, 1967.
- Yoshino, K. (1972). Models of choice behavior in two person non-zero sum games with response-dependent payoff matrices: Japanese Journal of Psychology Vol 43(1) Apr 1972, 31-40.
- Ziad, A. (1997). Pure-Strategy epsilon -Nash Equilibrium in n-Person Nonzero-Sum Discontinuous Games: Games and Economic Behavior Vol 20(2) Aug 1997, 238-249.

### DissertationsEdit

- Martin, R. J. (1974). Teaching children to cooperate: An experimental study using non-zero sum games: Dissertation Abstracts International.
- Steele, N. L. (2006). Social utility in zero-sum and non-zero-sum games, with and without competing norms. Dissertation Abstracts International: Section B: The Sciences and Engineering.

## External linksEdit

- Play zero-sum games online by Elmer G. Wiens.

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