Net Sum Zero Games (Definition + Examples)

Life often seems to resemble a type of game where there are losers and winners. People and whole societies often seem locked in business, war, political, and relationship games. If you have often wondered if there is a scientific way to reason about these ‘games,’ read on.

A net sum zero or zero-sum game is a highly competitive interaction where the winning participant’s gains are precisely equal to the loser’s losses. The overall net payoff is said to be zero because if you add the winners (+X) winnings to the loser’s (-X) losses, you get: +X – X = 0

Economists and social scientists study how humans compete for scarce resources and how they interact or what games they play to obtain them. A better understanding may lead to less destructive business and political decisions.

Zero-Sum Games And Game Theory

Zero-Sum Game is a term used in an applied branch of mathematics called Game Theory. ‘Game’ in this sense does not directly refer to recreational sports matches such as football, card games such as Poker, and board games such as monopoly.

Game Theory uses mathematics to model the rational decision options agents have in competitive situations, where agents need to be strategic because decisions are based on predicting the moves and reactions of opponents. This science of decision-making is now widely used in Economics and Social Science.

In 1921, the French mathematician, Emile Borel, wondered whether it was possible to mathematically model how competing agents such as Poker players can best bluff and second guess each other when very little information is available. He was curious to know if a ‘best’ strategy exists for any game.

The term Zero-Sum game was first used by Hungarian mathematician von Neumann in a mathematical paper written in German called “Zur Theorie der Gesellschaftsspiele,” which translates as “Theory of Parlor Games.” The paper was published in 1928 in the German mathematics journal: “Mathematische Annalen.” 

The game of Poker inspired Von Neumann. He realized that who won or lost did not depend on luck alone and hence could not be adequately modeled using probability theory alone. He was interested in how a rational player could skillfully counter an opponent’s best moves in a zero-sum game like Poker to minimize their losses under uncertainty.

In his paper, von Neumann introduced an optimum decision rule called MiniMax, which outlined the moves or decisions a rational player should make to minimize their losses in a “maximum loss,” or worst-case scenario, where the opponent is playing their best possible game. He formalized the notion of “bluff,” a strategy of deception and how best to hide information from opponents. Von Neuman further developed his ideas by teaming up with Oskar Morgenstern, an Austrian economist from Princeton. They published their book titled: Theory of Games and Economic Behavior in 1944.

Zero-Sum VS Non-Zero-Sum Games: Decisions And Interactions.

Many board games such as chess, card games such as Poker, and sports such as football are zero-sum games. These games have very well-defined rules and scoring systems, which make it clear who wins and who loses.

The scoring and point systems of board games and sporting contests are very well defined and make no allowances for the players’ subjective attitudes and psychological states. Therefore, even if losers gain more satisfaction or utility from playing than the winners, they still lose as defined by the rules.

Recreational games are played within well-defined boundaries and neatly isolated from life’s problems, which are not allowed to encroach on the playing field or game board. Because competitive behavior in real-life pursuits such as business, politics, and war are so often and poorly defined, it is challenging to identify zero-sum outcomes in these activities.

The random draw is the simplest zero-sum decision process used by people competing for scarce resources. For example: Imagine there are 2 brothers, John and Peter, who find a ticket to a Justin Bieber concert on a public park bench. Both brothers love Justin Beaver equally, but the ticket will allow only one entry.

Now, if they found a packet of chips instead, assuming the brothers are friends, it would be a win-win since they could share the chips equally. The ticket is, unfortunately, rendered useless if torn in half.

Since both brothers love Justin Beaver equally and will feel an equal sense of loss if denied the opportunity of going since they both found the ticket at the same time have an equal right to go, how do they fairly decide who goes? The best way to adjudicate in a zero-sum, non-merit-based situation is through an unbiased random draw, such as by drawing straws or flipping a fair coin.

Deciding to flip a coin, John calls out heads, and Peter calls out tails. Assuming tails wins, we can technically only call this situation zero-sum if John’s loss precisely cancels out Peter’s happiness. Now let’s assume that we could measure happiness in units of satisfaction called utils (from utility) and measure their happiness levels before finding the ticket as 10 Utils.

Assume that Peter’s and John’s satisfaction levels have been measured to increase by 5 Utils when attending a Justin Bieber concert. John and Peter’s coin flip only has a zero-sum outcome if Peter’s happiness increase is symmetrical (+5Utils) with John’s unhappiness or decrease in happiness (-5Utils).

Only if Peter is now enjoying 15 Utils at the concert, while John is sulking at home, sustained by only 5 Utils, can we talk of the outcome as being zero-sum. This is true because John’s loss is precisely canceled out by Peter’s gain, making no difference to the total Utils enjoyed.

Since 10utils +10Utils (before) = 15Utils + 5 Utils (after), the total satisfaction has been conserved and there hasn’t been a net change because the net change cancels to zero because +5Utils – 5Utils = 0Utils.

What if Peter feels guilty and so strongly empathizes with his brother that he tears up the ticket so that neither of them goes? How does this affect the total happiness score now? Will the total Utils, still be conserved at 20?

Whether total happiness will be conserved depends on the brothers’ psychological profile. It may very well be that Peter’s happiness at the concert will be precisely canceled by his guilt, but what happens to the collective happiness total depends on John’s reaction. John may think flipping the coin was fair and that Peter is silly to let the ticket go to waste.

John may validly ask Peter if he felt so guilty, why did he not give him the ticket instead? Now, none of them can benefit. If John now makes Peter realize his guilt is silly, he may regret tearing the ticket. The total happiness could quickly plummet because of Peter’s embarrassment for being silly and John’s resentment due to the wasted ticket.

Since the requirement that satisfaction must perfectly cancel dissatisfaction is highly improbable, John and Peter are likelier to experience a non-zero-sum outcome.

The perfect symmetry between John’s loss and Peter’s gain is a very special and unique case. In contrast, there are many non-unique ways in which John’s loss is non-symmetrical to Peter’s gain. For example, John may only ‘like’ Justin Beaver (+1Util), but Peter may be a hardcore fan (+5Utils).

If John ‘won’ the coin flip and went to the concert instead of Peter, the game would have a net negative-sum outcome. By going, John would increase his happiness by only 1 Util, while Peter would suffer a loss of 5utils. If the brothers have a close and friendly relationship, John would empathize with Peter, not insist on the coin flip and just give him the ticket.

Being wise, John may realize that cooperating, rather than competing with his brother and losing 1Util, so that his brother may gain 5 will result in more happiness overall since John’s Utils (10 – 1) added to Peter’s Utils (10 + 5) = 24Utils. That’s 4Utils above the present 20, making it a netpositive-sum game.

In contrast, if John decided to play a more competitive game, the brothers would collectively lose happiness: (10 + 1) + (10 – 5) = 16Utils, -4 from 20. Suppose John and Peter were strangers instead of brothers, and John knew he would never run into Peter again. In that case, he may decide to gain 1Util at Peter’s expense.

Since Peter and John are brothers and run into each other daily, John may reason that Peter may hold a grudge against him if he went, which may not be in his long-term interest. Sometimes in the future, John may need Peter’s cooperation and support.

This is why true zero-sum games are mostly played amongst strangers and in situations where the stakes are quantifiable, such as in money. If that is true, why are many fun children’s games, such as tic-tac-toe and rock-paper-scissor, zero-sum, played amongst friends and siblings?

While it is true that these games are Zero-sum in the sense that there is always a clear winner and loser (except in tic-tack-toe, where there is often a draw), children’s games are never played for high stakes, such as for money or property. They are played purely for fun, where the zero-sum competitive element merely enhances the thrill.

So long as both loser and winner have fun, the outcomes are a win-win, and the’ loser’ wins too, in a real sense and loses only in a formal sense. In games of skill such as chess, weaker players who lose against stronger players may see it as a win if they learned something new, thereby improving their game.

When adults play games such as Bridge and Poker for high financial stakes, then winning and losing is not just a formality, and the zero-sum nature of the game has very real consequences.

Is Business Competition Always A Zero-Sum Game?

When Businesses compete for a larger share of a small limited market, then outcomes are generally zero-sum if a business can only increase their market share at the expense of their rivals. However, if only a few small firms are selling in a market vast enough to absorb their collective maximum output, then outcomes could be win-win and positive-sum.

Once, Businesses start playing the ‘compete for the same customer’ game; it is not always clear whether the outcome will be zero-sum. Businesses can lure customers away from their rivals by decreasing their prices or through non-price factors such as brand differentiation, offering better quality, or improving customer service.

Suppose firms choose to compete by lowering prices. In that case, the outcome will be zero-sum, only if at least one of the firms is operating at an economy of scale, with low unit costs and profit margins or capital reserves sufficiently high to absorb the price war. At the same time, the other firms must be too weak to absorb it and exit the market.

However, suppose the rival firms miscalculate, with none having a clear scale or cost advantage. In that case, they could bankrupt each other, and the price war could turn out to be a lose-lose and a negative-sum game. To avoid a costly price war, many firms compete regarding non-price factors.

If firms skillfully identify and target different market segments, then there need not be a loser, and they all may be playing a win-win business game instead. For example, firm A may target low-income customers by offering low prices but cut costs by providing no after-purchase service, warranties, or delivery service.

Firm B may target higher income, more discerning customers by offering them higher quality and better customer service, but at higher prices. By serving different customer profiles, Firms A and B are not competing for the same customers, hence are playing a positive-sum rather than a zero-sum game.

Conclusion

Many board games such as chess, card games such as Poker, and sports such as football are neatly isolated from real life by the board and playing field boundaries. They are regarded as a recreational “time out” from the complexity and stress of real life.

Zero-sum outcomes are easily identifiable in well-defined recreational games. However, they are less identifiable in real-life competitive behaviors, such as wars, business, and politics.

References

https://en.wikipedia.org/wiki/Minimax_theorem

https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/neumann.html

https://en.wikipedia.org/wiki/Zero-sum_game

https://www.quora.com/Who-invented-the-term-zero-sum-game

https://study.com/academy/lesson/game-theory-positive-negative-zero-sum-games.html

Theodore T.

Theodore is a professional psychology educator with over 10 years of experience creating educational content on the internet. PracticalPsychology started as a helpful collection of psychological articles to help other students, which has expanded to a Youtube channel with over 2,000,000 subscribers and an online website with 500+ posts.