Back to main page

Back to home page

Printer friendly

Zero-Sum Games

Parking lots and poker

Someone once told me a story of a high school that had a problem that its parking lot wasn't big enough to accommodate all the students who wanted to park their cars there. In response to the complaints, the principal said "If you get here early enough, there are always plenty of spaces!" The trouble is, that, by the time school starts, if there are 500 spaces and 600 cars, it doesn't matter when the cars got there, there still will be 100 cars without spaces. The advice to come early might be helpful to an individual driver, but only at the expense of someone else who otherwise would have gotten that space.

Consider a group of friends who get together to play poker for money. Any individual might win, but only if someone else loses. If we add up all the money that was won by the winners and subtract what was lost by the losers, the total is always zero (we don't count the money spent on pizza as losing). That is, the sum of all the winnings, with losses counted as negative winnings, is zero. Mathematicians who work with game theory call this a zero-sum game. We might note that if the sum of the winnings is zero, the average amount won is also zero.

The nature of zero-sum games

Most of the implications are the same even if the sum isn't actually zero. If the sum of the prizes in a competitive game is significant, it is still the case that one person's gain is another person's loss, so effort applied on behalf of one participant doesn't help the group as a whole. Our concern here could more accurately be called "constant-sum games." Of course, even though this is part of "game" theory, we are not just interested in games but in how it can be applied to real-life situations.

It turns out that this kind of situation comes up frequently but is often unrecognized. People will spend a lot of effort devising a strategy that will help themselves or some group they are in, but if the only way they can gain is at the expense of someone else (possibly us) then we have to ask whether we should encourage this sort of policy. If we are selfish, it makes sense to put effort into doing well in fixed-sum situations. It also makes sense in sporting competitions, where the entertainment value and the learning that comes from competing provide a positive gain even though the prize or win/loss total is fixed. On the other hand, if we're supposed to be working for people's overall benefit, like the case of the principal with the parking spaces, it makes no sense to advocate strategies that only help some people at the expense of others.

Chain letters

From time to time we may get what is called a chain letter - nowadays it is more likely to be an email. These have created enough of a problem that they are illegal in most places, but we are still likely to encounter them from time to time. Typically it goes something like this: "There is a list of names and addresses below. Send $10 to the person at the top of the list and cross them off. Add your own name to the bottom of the list and send copies of the letter to five of your friends. If nobody breaks the chain, in a few weeks when your name is at the top of the list you should be getting $6250 from the 625 people who have letters with your name at the top." This would be the hypothetical payoff if there were four names on the list. It seems like you should get rich, but an obvious characteristic of this gimmick is that it is a zero-sum game. For every ten dollars I get, somebody must have spent ten dollars. The idea that everybody in the world could get rich this way cannot work. What happens in practice is that before long the community gets saturated with these things and people no longer keep sending them. For every person who makes $6250, there are 625 people who lose $10.

One tip that something is a zero-sum situation is that there is no useful work being done. If we make something or fix something or plan something or solve some problem, we are hopefully doing something that has benefit to people, so we have reason to think society as a whole is better off. If we are getting money for doing something useless, as in the case of the chain letter, we must be getting it at somebody else's expense.

College admissions

At one time I was an instructor at a college, and the administration sent a survey around to the faculty asking our opinions on what could be done to improve the school. One of the choices I picked was to try to recruit better students. This seemed sensible - the better the students, the more we could teach them and they would, on the average, be more successful after graduating. Some time later I thought about this and realized this had the quality of a zero-sum game. There is a certain part of the population that wants to go to college. Some are great students, some are not so great. But there's nothing the admissions or public relations departments could have done to make individual students better. If we got better students, that meant other colleges got worse students. Getting better students might have benefited our college, but it did not imply that society as a whole was any better off. If we improved our teaching methods or the curriculum or faculty selection, these could result in a net gain to education, but getting better students was just a rearrangement of who went where that would have no overall social benefit.

Athletic programs

At one time I was on a business trip to the Minneapolis area and happened to watch the local news. One story was that the University of Minnesota was embarking on a program to bring their athletic program up to top national standards. They would pay coaches more, do more vigorous recruiting, improve athletic facilities, and so on. This time I caught on more quickly. This was a zero-sum game. Within the Big Ten Conference, which Minnesota belonged to, the number of wins would still equal the number of losses. That would still be true for sports as a whole. Any extra games that Minnesota won because of this policy would be the result of other school losing more games. If every college pumped up their programs equally, each would spend a lot more money, and nobody, on the average, would be any better off in terms of winning. From the standpoint of spending money, the overall total would be a substantial loss.

Fighting crime

Around 1990 in downtown Minneapolis there was an area known as "E block" that was notorious for drug dealing and prostitution. Disreputable people hung around the area looking to do business with people who patronized the bars and shops there. The city decided to attack the problem by buying up the businesses on E block and tearing them down. The area was made into a parking lot. Sure enough, the dealers and prostitutes stopped doing business on that block. This strategy bothered me, however, since there seemed to be no reason to expect that the undesirable people who had worked the area weren't just doing the same things somewhere else. The buildings were eliminated, but the buildings weren't committing any crimes. The people who were committing the crimes were still somewhere, presumably in the same numbers and doing the same things.

It's possible that the presence of those buildings or the businesses going on there made the criminal activities easier to commit or harder to prevent, but at no point did I hear any politician or news reporter make that case. As far as I could tell it was simply a matter that since these things were no longer happening on E block, the city was somehow better off. My own feeling is that it might have been better for police to keep an eye on what was happening if these activities tended to be concentrated in a known area. I did notice that the overall crime rate in Minneapolis was just as bad following the cleanup as it was before. It seems like this was another case of improving a particular highly visible area at the expense of other less visible areas. The crime in E block decreased, but overall crime did not.

A few years later there were news stories about a lot of drug activity taking place at a particular apartment building. Perhaps encouraged by the "success" at E block, the city bought the building and razed it.

Economic situations

Game theory is most often applied to economics, and this is an area where being aware of zero-sum games can be particularly useful. The acronym TANSTAAFL is sometimes heard in economic circles. It stands for "There Ain't No Such Thing As A Free Lunch". What it means is that when something seems to be free, there has to be somebody somewhere who is paying for it, and it may turn out to be you. The term is probably most often used with respect to government and taxes. If a government program promises to give you some financial assistance or free service, somebody, typically the taxpayer, has to be paying for it. This can happen with business as well. When you use coupons to get something free or at a reduced price, or get a free gift, or you take advantage of some other promotion a business puts on, somebody pays for it. Normally this has to be paid for out of higher prices for the products or services you get from that company. If not, the company would go out of business. Unless the apparent windfall requires you or someone else to do some productive work, the overall sum of goods and services is not changed, so your gain has to come from someplace. We would like to imagine that it comes from wealthy investors or corporate management, but that is not likely to be the case.

A huge activity that has to be paid for is advertising and sponsorships. When we watch "free" television, these are expensive productions that somebody is paying for. Obviously the people who put on the commercials are footing this bill. The commercials themselves can also be quite expensive. All this expense has to be paid for though markup on the products or services being sold. It is something to keep in mind when a commercial makes some product seem appealing.


A business that has an important zero-sum component is insurance. We pay money in the form of premiums to insurance companies to guard against a variety of situations such as premature death, disability, expensive medical treatments, auto accidents, house fires, theft, and so on. Undoubtedly we buy a certain peace of mind knowing that we (or our family) will be able to survive financially if one of these events occurs. On the other hand, we have to recognize that the company has no real source of revenue except premiums, and so all the money they pay out, plus their profits and expenses, must come from what is paid in. In overall terms, the customers are paying more than what they get back. There are a few complicating factors. One is that the insurance company invests the premiums, so they can get some return on investment in addition to the premiums themselves. However any money we pay in premiums is money that we cannot invest ourselves, so there isn't much difference here. Sometimes employers pay part of the premium as an employee benefit, so we may come out ahead on the average in this kind of situation.

On the other hand there are expenses the insurance company has that we wouldn't have if we paid bills ourselves. One is that it is common for people like dentists and auto repair shops to charge more if they know insurance is paying the bill than they do if we are not insured. In addition there are people who fake injuries or accidents to cheat insurance companies, and there are costs for investigation to try to keep such cheating at a minimum. For the most part, the money we get back from insurance is considerably less than what we pay in. This is particularly true for things like service contracts on appliances or electronic devices. A sensible policy is probably to only insure against major calamities that would otherwise create serious financial hardship and to put the money that would have gone into premiums into savings or investments. On the things you do insure, pick plans with high deductibles that have lower premiums. You will then be able to pay for the minor expenses from your savings and probably have a reasonable amount left over.

I suspect most people overestimate the likelihood that they will collect on insurance, and so the think it is a better deal than it is. But recognizing that it is like a zero-sum game - all of the money paid out ultimately comes from our premiums - should make it clear that, on the average, we lose on this kind of bet (casinos work the same way but don't give you any peace of mind).

Summing up

Here are some other constant-sum situations: people competing for class rank in a school, people competing to make a good impression in a job interview, trying to time stock market transactions to make money, acquiring status symbols so you seem more successful than your neighbors, trying to win as much money as possible in lawsuits and divorce settlements, cities trying to attract businesses or sports teams to the area, and local governments trying to get undesirable people to move somewhere else.

If we provide some useful work and that earns us money, we are not engaged in a zero-sum game and both we and the people we interact with can come out ahead. If we get money without doing anything useful, then any success we have comes at someone else's expense, and we should be aware that our success depends not merely on doing the right thing, but on doing it better than the people we are competing against. I have found that being aware of zero-sum situations is very useful in understanding the nature of many complex situations and whether we can realistically expect to gain from some kinds of transactions.