# Can You Solve The Lions And Lambs Classic Game Theory Puzzle?

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.

## The solution

This type of game theory problem, where you need to find a solution for a general value of N (where N is a positive whole number), is a good way of testing game theorists’ logic and of demonstrating how backward induction works. Logical induction involves using evidence to form a conclusion that is probably true. Backward induction is a way of finding a well-defined answer to a problem by going back, step-by-step, to the very basic case, which can be solved by a simple logical argument.

## Get The Latest From InnerSelf

In the lions game, the basic case would be N=1. If there was only one hungry lion on the island it would not hesitate to eat the lamb, since there are no other lions to compete with it.

Now let’s see what happens in the case of N=2. Both lions conclude that if one of them eats the lamb and becomes too full to defend itself, it would be eaten by the other lion. As a result, neither of the two would attempt to eat the lamb and all three animals would live happily together eating grass on the island (if living a life solely dependent on the rationality of two hungry lions can be called happy).

For N=3, if any one of the lions eats the lamb (effectively becoming a defenceless lamb itself), it would reduce the game to the same scenario as for N=2, in which neither of the remaining lions will attempt to consume the newly defenceless lion. So the lion that is closest to the actual lamb, eats it and three lions remain on the island without attempting to murder each other.

And for N=4, if any of the lions eat the lamb, it would reduce the game to the N=3 scenario, which would mean that the lion that ate the lamb would end up being eaten itself. As none of the lions want that to happen, they leave the lamb alone.

Essentially, the outcome of the game is decided by the action of the lion closest to the lamb. For each integer N, the lion realises that eating the lamb would reduce the game to the case of N-1. If the N-1 case results in the survival of the lamb, the closest lion eats it. Otherwise, all the lions let the lamb live. So, following the logic back to the base case every time, we can conclude that the lamb will always be eaten when N is an odd number and will survive when N is an even number.

Amirlan Seksenbayev, PhD Candidate in Mathematical Sciences, Probability and Applications, Queen Mary University of London

### Related Books

{amazonWS:searchindex=Books;keywords=game theory;maxresults=3}

## You May Also Like

Get The Latest By Email

{emailcloak=off}

### INNERSELF VOICES

###### How We've Created Duality and Separation, and What To Do About It
by Judith Corvin-Blackburn
by Jim Willis
by Dery Dyer
by Paul Selig
by Pam Younghans
###### Expansion: Growing Beyond Limits and Stepping Outside Your Comfort Zone
by Terri-Ann Russell
###### Glad to Be Human: Finding Numberless Reasons To Feel Grateful and Hopeful
by Irene O'Garden
###### Remembering Who We Are Destined To Be and Letting What No Longer Serves Us "Die"
by Jennifer T. Gehl
###### The Elephant in the Room: You Can Ignore Him But He's Still There
by Jane Duncan Rogers
by Jean Walters
by Alan Cohen
###### I Am Safe! You Are Safe! We Are Safe!
by Marie T. Russell

###### Glad to Be Human: Finding Numberless Reasons To Feel Grateful and Hopeful
by Irene O'Garden
by Jean Walters
###### Expansion: Growing Beyond Limits and Stepping Outside Your Comfort Zone
by Terri-Ann Russell
###### How To Keep Your Home Workspace Safe And Hygienic
by Libby Sander, Lotti Tajouri, and Rashed Alghafri
###### Remembering Who We Are Destined To Be and Letting What No Longer Serves Us "Die"
by Jennifer T. Gehl
###### How We've Created Duality and Separation, and What To Do About It
by Judith Corvin-Blackburn
by Agnes Mueller
by William Petri
###### The Elephant in the Room: You Can Ignore Him But He's Still There
by Jane Duncan Rogers
###### These Dogs Are Trained To Sniff Out The Coronavirus. Most Have A 100% Success Rate
by Susan Hazel and Anne-Lise Chaber
by U. Chicago
by Pat Harriman
by Irene Gammel

### FROM THE EDITORS

###### The Day Of Reckoning Has Come For The GOP
by Robert Jennings, InnerSelf.com
The Republican party is no longer a pro-America political party. It is an illegitimate pseudo-political party full of radicals and reactionaries whose stated goal is to disrupt, destabilize, and…
###### Why Donald Trump Could Be History's Biggest Loser
by Robert Jennings, InnerSelf.com
Updated July 2, 20020 - This whole coronavirus pandemic is costing a fortune, maybe 2 or 3 or 4 fortunes, all of unknown size. Oh yeah, and, hundreds of thousands, maybe a million, of people will die…
###### Blue-Eyes vs Brown Eyes: How Racism is Taught
by Marie T. Russell, InnerSelf
In this 1992 Oprah Show episode, award-winning anti-racism activist and educator Jane Elliott taught the audience a tough lesson about racism by demonstrating just how easy it is to learn prejudice.
###### A Change Is Gonna Come...
by Marie T. Russell, InnerSelf
(May 30, 2020) As I watch the news on the events in Philadephia and other cities in the country, my heart aches for what is transpiring. I know that this is part of the greater change that is taking…
###### A Song Can Uplift the Heart and Soul
by Marie T. Russell, InnerSelf
I have several ways that I use to clear the darkness from my mind when I find it has crept in. One is gardening, or spending time in nature. The other is silence. Another way is reading. And one that…