Of Gains and Losses

An Introduction to Zero-Sum Situations

Shivamjha

Published in

Intellectually Yours

6 min readFeb 23, 2021

“It’s not a question of enough…It’s a zero-sum game. Somebody wins. Somebody loses.” – Gordon Gecko, Wall Street

A poker table — Poker is a classic example of a Zero-Sum Game.

A zero-sum situation is a situation in Game Theory wherein one individual’s benefit is equal to another’s deficit, so the net change in riches or profits is zero. A zero-sum situation is not limited to only two players. In economic markets, choices and prospects are instances of zero-sum situations, barring transaction costs. For each individual who gains on an agreement, there is a counterparty who loses.

Understanding Zero-Sum Games

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 equal 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 short, where nothing new is being created or nothing is taken out of the system i.e. only exchanges happen in terms of profit and losses as known as zero-sum games.

For example, a two-player game where the order of play proceeds as follows:

The first player (red) chooses one of the actions (1 or 2) without the other player knowing what he chose. Then the second player (blue) chooses one of the actions (A, B, or C) without the other player knowing what he chose. After both players have their respective choices, the scores are revealed according to the payoff table given below.

Let’s take a case where Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

In this particular game, both players know the payoff matrix and attempt to maximize their total number of points. Red could reason as follows: “With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better.” With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red’s reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.

Now that we have established what zero-sum situations really are, let’s look at how these are different from many well-known examples.

Zero-sum situations are totally opposite to win-win situations, for example, an economic alliance that essentially expands exchange between the two countries where both sides have a profit. — or lose-lose situations, similar to wars, where both sides face huge losses of life and resources.

Another very popular example of a non-zero-sum is The Prisoner’s Dilemma, where there is an unequal distribution of profits and losses.

Applications of Zero-Sum Games

The minimax theorem:

This theorem can be understood very simply through zero-sum games. For instance, let there be a two-player zero-sum game: Player A knows that if she maximizes her return from the game, she must minimize that of her opponent, player B. The outcomes can be mathematically graphed, ranging from total wins by A or B to a situation where A and B find an equilibrium point, from which they can’t rationally deviate without harming their own returns.

In other terms, the more A wins, the more B loses. It is also notable that Nash Equilibrium was proved with the help of this theorem.

The graph of minimax theorem; f(x,y)=x²-y²

Nash equilibrium:

Nash was dealing with a zero-sum game, but with added complexities: Both players have their own strategy and know the strategy of their opponent. An equilibrium is reached when neither player can benefit from changing their strategies if their opponent’s scheme remains unchanged; thus, you can predict the outcome of a zero-sum game by understanding the strategy of just one player.

Zero-sum games in stocks:

Investing in stocks is a zero-sum game because the outperformance of one investment strategy requires the underperformance of other strategies. There will always be a buyer and a seller of a stock, and the gain of one will involve a loss for the other. Investors’ collective performance in the stock market relative to an index over a fixed period of time (usually a day) can also be viewed as a zero-sum game. Since the value of an index includes all gains and losses, it is by definition zero-sum.

One’s “gain” (buyer) requires another’s “loss”(seller) in the stock market; Photo by Sigmund on Unsplash

Real-life Examples of Zero-Sum Games

Diplomacy in International relations:

If we take a close look at the current situation of India and Pakistan, we will realise that hawks see the relationship as a zero-sum game while the doves see the same relation as a non zero game, more precisely a win-win situation.
Hawks see that every decision made results in either the victory of their side or the opponent’s side, thus they are very reluctant to cooperate and make concessions. While on the other hand, doves are more than willing to cooperate and reach an agreement resulting in a win-win situation and avoiding conflict.

Biology

In many host-parasite relations, the parasite has numerous variants, antigenic strains, or types. The host also has many types of reactions and defenses. We here see that it is a zero-sum situation where in order to maximise the gains of the host, it has to prevent the parasite from acting and vice versa.

However, in the case of symbiotic relations, we see that it is a non-zero-sum situation. Take the example of bees gathering nectar from flowers, which they make into food. Pollen rubs onto their bodies as they collect the nectar, and the pollen then falls off into the next flower, which pollinates it. We see that this is a win/win situation for both parties.

Some Misconceptions of Zero-Sum situations

For example, elections are not zero-sum games because the pool of people who vote is not fixed. We may know how many people are eligible to vote, but we can never know in advance the exact number who will actually cast ballots. It can happen and often does that a candidate will gain votes in a second electoral round without her opponent necessarily losing votes.
This is because new voters appeared who did not participate in the first round. Campaign tactics can offer a pretty good idea of how candidates view an election. If they run negative political campaigns, it’s a clue that they consider elections to be a zero-sum game: They try to put down their opponents because they strongly associate the opponent’s loss with their gain. What these candidates may not consider is that voters can choose not to vote.

Another example can be that of a professional golf match between legends: Tiger Woods vs Rory Mcllroy

Sometimes the notion of winner and loser isn’t quite clear. If Tiger Woods gets into a playoff with Rory McIlroy in the Master’s Tournament, for playing four days of golf, one of them will take home $1.5 million while the other gets $1 million. Here even before playing a single swing, they are guaranteed prize money of $1 million.

Takeaway?

In our day-to-day lives, we often encounter these types of win/win situations which on the surface, look like a zero-sum game, a win/lose situation. Game Theory teaches us to identify gains and losses, and the very notion of a “gain” and a “loss”. Now equipped with this knowledge, can you change your pay-offs to turn every zero-sum game into a non-zero-sum game?