The boy who cried wolf

The boy who cried wolf

David Huang
Game Theory
April 6, 2014

Once upon a time, a boy decides that his microeconomics class taught by Jeff Borland isn’t worth his time and instead, settles on a career as a shepherd. However, his career choice is monotonous and he finds happiness in fooling the local village into believing there is a wolf at large. The rest of this classic story results in all of his sheep being eaten, as you know.

Alas, had this boy stuck with his classes and learnt game theory, this could have been prevented.

In a decision tree, we map out results of this game and assign arbitrary numbers to its payoffs and probabilities. We assume for simplicity that the villagers have no choice but to either help him if he is not a liar and abandon him if he is. Nature (or random chance) draws the boy either a ‘Wolf’ with 10 per cent probability or ‘Nothing’ with 90 per cent probability and to simplify things, we assume the first period draws ‘Nothing’. Either way, the boy has choices of ‘Cry’ or ‘Silence’ at the second and fourth nodes. Let zero be the payoff he receives for being saved if a wolf appears and he plays ‘Cry’. Similarly, if he keeps silent when no wolf arrives, he is no better or worse off, thus receiving a payoff of zero in these cases too.


If nature decides on no wolf appearing both times and the boy decides on fooling the villagers at the second node, then let’s say he receives a utility of five from the amusement of watching the villagers run around looking for a wolf. Crucially, we assume that the villagers’ goodwill is limited to a single period so in the next period, regardless if a wolf comes or not, the boy loses the ability to call out the villagers so his fate is fixed to whatever hand nature deals him. The red line marks the path of the boy in our fable, where he lies in the first period but then in the second period, draws a ‘Wolf’ and realises he is doomed no matter what action he takes. We hence assign a payoff of negative 100 as the worst case scenario in our game.


Assuming that he will only make a rational choice at node four, he will choose the lines marked in blue. We use the theory of backward induction to see if in the first period he should choose ‘Cry’ or ‘Silence’. Backward induction is the process of considering first the possible end results to determine the decision you should undertake today. Because there are probabilities involved here, we are required to use expected utility.

Therefore, for the boy:

The expected return of playing “Cry” period 1

The expected return of playing “Silence” period 1

To summarise, we see that the boy would opt to be truthful in the first place as this would yield an expected utility of -5.5. It is useful to note that depending on the assigned probabilities, the results to our game would not always hold. Regardless, this simple model teaches us an important life lesson:

Reputation is important.

Now think about the same model, but extended over an infinite number of periods. We would find that it is in the boy’s interests to choose to tell the truth at each node, as one lie would mean that the villagers would leave him at the mercy of nature – a ‘grim trigger’ strategy [1]. Some might suggest that at the final node, the boy should always choose to fool the villagers but since we never reach the last period here, as a rational being he should choose to maintain a clean reputation so the villagers recognise his good faith, and thus would be willing to help him.

Other applications

In poker, for instance, game theory makes up a major component of the game [2]. You take into account how you wish the other players to perceive you, thus manipulating their possible future actions by adjusting your current actions accordingly. You may wish to initially establish yourself as a consistent bluffer only for you to later flip your cards and reveal your powerhouse hand.

In pop culture, strategy and game theory can be associated with some interesting characters. Severus Snape establishes himself as a malicious ‘Death Eater’ to obtain the opportunity to save our protagonist’s life at a ‘final decision node’, altering the Harry Potter universe. In the movie ‘The Usual Suspects’, Kevin Spacey portrays himself a cripple under the perfect guise of ‘Verbal’; physically and also mentally weakened. However, as we learn in the final act of the movie, his artificial reputation could not be further from the truth.

Additionally, Blackbeard was a legendary pirate who wished to avoid fights by cultivating a fearsome reputation that has survived until today. Trending a large black coat and placing a slow-burning fuse in his long black hair, he instilled a supernatural terror that led many to surrender without a fight – an optimal outcome for Blackbeard as it meant ships and their loot were left intact [3].

In economic terms, ‘signalling’ is identified as one party selectively feeding information to another party with the aim of positioning them to behave in a certain way in an attempt to obtain a desired result.

Utilising game theory, we should be able answer the following questions:

What should I wear today?

What things should or shouldn’t I say to her?

What should I do about this situation?

Obviously there are no easy answers to these questions, nor is there necessarily a certain strategy involved in any decision affecting reputation. It is worthwhile to remember simply that the actions we take today could potentially come back to haunt or benefit us later. In the case of Aesop’s Tale, if the boy wished for others to respond to him in kindness then he too should have treated others fairly by being truthful.

[2] Swanson, J. (2005). Game Theory and Poker.