By Mary Patterson
I remember my daughter’s first field day at school. It was a sunny beautiful day, the children were running relays in teams, playing tug of war, and throwing water balloons. When the day was over, I was so excited to see which team had won. Instead, the gym teacher handed out individual awards for best attitude and best sportsmanship.
“What?” I felt my blood surge, “No winners? No second place ribbons? No losers? What is this?” As an American, I was taught that competition meant winners and losers. In game theory, when there is a winner and a loser, this is called a zero sum game. Someone wins, +, and someone loses, -, and these add up to zero. Chess, monopoly, and poker are all zero sum games because someone wins and someone always loses.
A different type of game, however, became famous with John Nash’s first paper on bargaining. John Nash was the subject of the film, “A Beautiful Mind.” His first paper, conceived when he was just 17, discussed the non-zero sum game. In this type of game both players win, +, +, and therefore the results add up to more than zero. When I was first reading about non-zero sum games I asked my daughter if she knew of any. “Charades,” she replied quickly. In a simple game of charades, if the person miming the word performs well, and the guesser performs well, then the word is discovered, win, win, +,+. See-saw and playing catch are also examples of non-zero sum games.
Non-zero sum game in real life
In economics, auctions such as Ebay would be considered a non-zero sum transaction. If we buy something at the store, then usually we pay a surplus (we lose, the store wins) or we buy something on clearance (we win, the store loses). But, on Ebay, an optimal price is reached due to the bidding process. The buyer gets the correct price and the seller gets the real value for his/her product, win, win,+,+.
In international politics, war is usually considered to be a zero sum game, someone wins,+, and someone loses, -. Many would argue that war is actually a negative number game where both parties lose, -,-. There is a tremendous loss of life and also economic chaos during war, and optimal outcomes are rarely reached. John Nash provided a mathematical model that would determine the optimum win, win situations in economics, biology and international politics.
The Nash Equilibrium: Examples
Let’s look at an example of optimal strategy in game theory. Imagine that there are several hungry ducks in a lake. Let’s say we put one person at the north end of the lake. This person throws small square pieces of bread into the lake at a rate of 1 piece every 5 seconds. Another person stands at the south end of the lake, and throws the same size of bread, but, only throws 1 piece every 10 seconds. Now, if you are a duck, where do you swim? If you swim to the faster rate of bread, chances are that all of the other ducks will think the same way and you will not get any bread. Using Nash’s mathematics, it can be determined that for each duck to receive the optimal amount of bread, 2/3 of the ducks should go to the faster bread thrower, and 1/3 of the ducks should go to the slower bread thrower. This is what is called a Nash Equilibrium.
A Nash Equilibrium is the set of strategies that are optimal for each party involved in the transaction. Amazingly, the ducks figure this out in about one minute, 1/3 of them go to the slower bread thrower and 2/3 go to the faster bread thrower.
The following is a classic game theory problem. Two men go hunting for deer. If they cooperate, then they will get the deer. If they don’t cooperate their chances of getting the deer are greatly diminished. But, if they individually chase the rabbit, then one of them will get the rabbit. So, each hunter has a decision to make. If he goes for the deer, there is a chance that the other hunter will not help him, and he will have less chance of success. If he goes for the rabbit, he will definitely get the rabbit, but, that is less than the deer. If both men go for the deer, then they will gain the most out of their efforts.
John Nash showed that if the two hunters do not communicate, that the Nash Equilibrium, the best strategy to adopt is to go for the rabbit. (To see chart, go here.) This is the best individual strategy if they do not know what their partner will do. However, this would not be the best endpoint for the two hunters. If they communicate with each other, then they can both hunt the deer and be successful.
Applications for peace
In peace negotiations, the purpose of the negotiator is to convince the players to participate in the option most beneficial to both parties. Without communication, or someone to negotiate, the best strategy would be to chase the rabbit, even though this is not the optimal outcome.
Israel and Egypt had problems with the Sinai Desert. Both sides wanted the land. The Israelis wanted the security that the space of the land would offer, while Egypt wanted to maintain its territorial claims. Former President Jimmy Carter led negotiations of a peace treaty and both sides were convinced that sovereignty and security could coexist. Presidents Sadat and Begin would not even speak to each other during most of the negotiations. Jimmy Carter had to shuttle between rooms to convince both leaders of optimal win-win strategies. Egypt committed to the non-militarization of the the Sinai desert and Israel withdrew its troops. If a war had ensued, thousands of people would have been killed and it is possible that neither of the desired outcomes would have been achieved.
The current situation between the U.S. and Iran seems to be worsening. The U.S. wants Iran to stop enriching Uranium, and Iran wants the U.S. to leave Iraq. There are a myriad of other issues on the table, of course, but, these are the main issues. With insufficient communication, war might result, because each party is suspicious of the strategy of the other. John Nash provided mathematical models (much too complicated to go into here) that could deal with multiple parties and strategies, he proved that an optimal strategy could be adopted.
The problem with a pre-emptive strike against Iran is that it would be nearly impossible to predict the plausible outcomes of such a move. If we attack Iran, how could we ensure that they would remain nuclear free? Nuclear weapons can easily cross borders and might come from other countries following a war. Also, it is impossible to predict the loss of life, the reaction of other nations, the economic costs, and the political price that war would entail. However, if we pursue negotiations with Iran, it might provide us with a viable means of removing ourselves from Iraq. Also, Iran may consider a cessation of their nuclear program in exchange for a military buffer zone. A win-win situation is possible for both countries, but, such a scenario is impossible without viable negotiations.
Teaching cooperation
Non-zero sum games provide a strong base on which the lessons of peace negotiations can be taught. We can expose children to the joys of cooperative play at early ages. Children can build towers together, build human pyramids, play catch, and other simple win-win games. Older children can be taught cooperative games, like the counting game. In this game the children stand in a circle and must count out loud to 20 without indicating verbally who will say the next number. If two people shout out a number simultaneously, they must start again.
Christians can ask their children to look for the non-zero sum games spoken of by Jesus, “Give unto Caesar what is Caesar’s, give unto God what is God’s,” (Jesus was a master of the win-win scenario). Islamic parents can ask their children to look for win-win verses in the Koran. And most religions contain the game theory of, “Do unto others as you would have them do unto you.”
It is pretty clear where the Nash Equilibrium is in that game. (To see a chart showing how, go here.)
Sources for this article came from books and websites listed below. They are recommended for further reading on non-zero sum games.
A Beautiful Math, John Nash, Game Theory, and the Quest for a Code of Nature
By Tom Siegfried Copyright 2006
The Essential John Nash
Edited by Harold W. Kuhn and Sylvia Nasare Copyright 2002
http://objectiveministries.org/gametheory/ by Jack Holgroth
http://www.nonzero.org/ by Robert Wrigh
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