How many lions does it take to kill a lamb? The answer isn’t as straightforward as you might think. Not, at least, according to game theory.
Game theory is a branch of maths that studies and predicts decision-making. It often involves creating hypothetical scenarios, or “games”, whereby a number of individuals called “players” or “agents” can choose from a defined set of actions according to a series of rules. Each action will have a “pay-off” and the aim is usually to find the maximum pay-off for each player in order to work out how they would likely behave.
This method has been used in a wide variety of subjects, including economics, biology, politics and psychology, and to help explain behaviour in auctions, voting and market competition. But game theory, thanks to its nature, has also given rise to some entertaining brain teasers.
One of the less famous of these puzzles involves working out how players will compete over resources, in this case hungry lions and a tasty lamb. A group of lions live on an island covered in grass but with no other animals. The lions are identical, perfectly rational and aware that all the others are rational. They are also aware that all the other lions are aware that all the others are rational, and so on. This mutual awareness is what’s referred to as “common knowledge”. It makes sure that no lion would take a chance or try to outsmart the others.
Naturally, the lions are extremely hungry but they do not attempt to fight each other because they are identical in physical strength and so would inevitably all end up dead. As they are all perfectly rational, each lion prefers a hungry life to a certain death. With no alternative, they can survive by eating an essentially unlimited supply of grass, but they would all prefer to consume something meatier.
One day, a lamb miraculously appears on the island. What an unfortunate creature it seems. Yet it actually has a chance of surviving this hell, depending on the number of lions (represented by the letter N). If any lion consumes the defenceless lamb, it will become too full to defend himself from the other lions.
Assuming that the lions cannot share, the challenge is to work out whether or not the lamb will survive depending on the value of N. Or, to put it another way, what is the best course of action for each lion – to eat the lamb or not eat the lamb – depending on how many others there are in the group.