Zero-Sum Game

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What is a 'Zero-Sum Game'

Zero-sum is a situation in game theory in which one person’s gain is equivalent to another’s loss, so the net change in wealth or benefit is zero. A zero-sum game may have as few as two players, or millions of participants.

Zero-sum games are found in game theory, but are less common than non-zero sum games. Poker and gambling are popular examples of zero-sum games since the sum of the amounts won by some players equals the combined losses of the others. Games like chess and tennis, where there is one winner and one loser, are also zero-sum games. In the financial markets, options and futures are examples of zero-sum games, excluding transaction costs. For every person who gains on a contract, there is a counter-party who loses.

BREAKING DOWN 'Zero-Sum Game'

In game theory, the game of matching pennies is often cited as an example of a zero-sum game. The game involves two players, A and B, simultaneously placing a penny on the table. The payoff depends on whether the pennies match or not. If both pennies are heads or tails, Player A wins and keeps Player B’s penny; if they do not match, Player B wins and keeps Player A’s penny.

This is a zero-sum game because one player’s gain is the other’s loss. The payoffs for Players A and B are shown in the table below, with the first numeral in cells (a) through (d) representing Player A’s payoff, and the second numeral Player B’s playoff. As can be seen, the combined playoff for A and B in all four cells is zero.

Zero Sum Game - Matching Pennies

Most other popular game theory strategies like the prisoner’s dilemma, Cournot Competition, Centipede Game and Deadlock are non-zero sum.

Zero-sum games are the opposite of win-win situations – such as a trade agreement that significantly increases trade between two nations – or lose-lose situations, like war for instance. In real life, however, things are not always so clear-cut, and gains and losses are often difficult to quantify.

In the stock market, trading is often thought of a zero-sum game. However, because trades are made on the basis of future expectations and traders have different preferences for risk, a trade can be mutually beneficial. Investing longer term is a positive sum situation because capital flows facilitation production and jobs that then provide production and jobs that then provide savings and income that then provide investment to continue the cycle.

History of Zero-Sum Game Theory

Game theory is a complex theoretical study in economics. The 1944 groundbreaking work “Theory of Games and Economic Behavior,” written by Hungarian-born American mathematician John von Neumann and co-written by Oskar Morgenstern, is the foundational text. Game theory is the study of strategic decision making between two or more intelligent and rational parties. The theory, when applied to economics, uses mathematical formulas and equations to predict outcomes in a transaction, taking into account many different factors, including gains, losses, optimality and individual behaviors.

Game theory can be used in a wide array of economic fields, including experimental economics, which uses experiments in a controlled setting to test economic theories with more real-world insight. In theory, zero-sum game is solved via three solutions, perhaps the most notable of which is the Nash Equilibrium, put forth by John Nash in his 1951 paper “Non-Cooperative Games.” The Nash equilibrium states that two or more opponents in the game, given knowledge of each others’ choices and that they will not receive any benefit from changing their choice, will therefore not deviate from their choice.

Zero-Sum Game and Economics

When applied specifically to economics there are multiple factors to consider when understanding a zero-sum game. Zero-sum game assumes a version of perfect competition and perfect information; that is, both opponents in the model have all the relevant information to make an informed decision. To take a step back, most transactions or trades are inherently non zero-sum games because when two parties agree to trade they do so with the understanding that the goods or services they are receiving are more valuable than the goods or services they are trading for it, after transaction costs. This is called positive-sum, and most transactions fall under this category.

Options and futures trading is the closest practical example to a zero-sum game scenario. Options and futures are essentially informed bets on what the future price of a certain commodity will be in a strict timeframe. While this is a very simplified explanation of options and futures, generally if the price of that commodity rises (usually against market expectations) within that timeframe, you can sell the futures contract at a profit. Thus, if an investor makes money off of that bet, there will be a corresponding loss. This is why futures and options trading often comes with disclaimers to not be undertaken by inexperienced traders. However, futures and options provide liquidity for the corresponding markets and can be very successful for the right investor or company.