Volunteer's Dilemma

Volunteer's Dilemma

It is recommended you read the blog post on the Prisoner's Dilemma before continuing.

You find yourself as a night guard to an ancient Roman village. Stationed at your inconspicuous post just outside the village boarder, you notice the Gauls (Rome's enemies back in the day) stealthily approach, armed to the teeth with weapons. Their intent is easy to guess: kill all Roman guardsmen and sack the village. Still, you and all of your fellow guards are stationed in hard-to-see spots. If you remain silent, the Gauls will walk right by you into the village. Sound the alarm to alert the townspeople and other guards, however, and they will descend upon your position like ants to spilled food. Here lies the dilemma. Each guard, you included, has to either make a sacrifice such that all others will benefit, or to do nothing.

	At least one other guard sounds the alarm	All other guards stay quiet
Sound the Alarm	Village safe, one guard dies, might be you	Village safe, you die
Stay quiet	Village safe, you live	All villagers die

Luckily for everyone, you and the rest of the guard have enough honor to believe that the safety of your village has greater value than each single guard's life. Still, you want to avoid the chance of sacrificing your life when another guard would have sounded the alarm for you. You can't know what the other guards are going to do, so you have to make the critical decision alone. If you guess another guard will sacrifice himself, the best option is to stay quiet. If you guess other guards will stay quiet, the best option is to sound the alarm. As such, there is no dominate strategy.
Very interesting to note is that, in general, people will guess that someone else will make the sacrifice the more other people there are. That is, if you are defending a village with 1000 soldiers, surely, you guess, one of them is going to make the sacrifice. Similarly, a defense of only you and one other guard might leave you hesitant to believe that the one other guard will make the sacrifice. Naturally, if it's just you, you can be sure that no one else has done anything. The kicker is that the strength of the guess about whether someone else has made the sacrifice is directly related to how many other people there are, and that your decision to stay silent is based on the strength of that guess. As a result, the more people in the scenario, the less likely any one person is going to make the sacrifice, giving this dilemma a paradoxical quality.

Here's the abstract payoff matrix for the Volunteer's Dilemma, using numbers to represent utility.

	Others sacrifice	Others defect
You sacrifice	You: 9, Others: 9	You: 9, Others: 10
You defect	You: 10, Others: 9	You: 0, Others: 0

There are two equilibria in the dilemma: the top right and bottom left positions. If the guess that someone else has made the sacrifice is strong enough, all parties are pushed towards defection, resulting in what is clearly the worst outcome.

Truly a noble man making a noble sacrifice.

The phenomenon where people are less likely to act when in the presence of others is strongly related to the bystander effect and the diffusion of responsibility. While it would be nice to change our nature, to realize that our own sacrifice is so trivial as to warrant being ignored in situations where the greater good is sizable, the truth is that this does not always happen. Our sacrifices are likely felt so personally as to be impossible to ignore, making the Volunteer's Dilemma and bystander effect a very real, and sometimes very tragic, event.

Sources and Further Reading:
Volunteer's Dilemma on Wikipedia
Bystander Effect on Wikipedia
Case study on the Volunteer's Dilemma