Volunteer's Dilemma

The Volunteer's Dilemma models a situation in which each player faces the decision of either making a small sacrifice from which all will benefit, or freeriding.

In other words, the choice each player has to make in the Volunteer's Dilemma is whether to "volunteer, or not volunteer" where volunteering comes at some cost, but benefits every player.

For example: Say that the electricity has gone out in Jencks.

Every student who lives in Jencks knows that campus security can fix the problem as long as one person calls to notify them, at some cost.

If no one volunteers, the residents of Jencks get the worst possible outcome: no lights, no cooking, no laundry.

However, if one student volunteers to call security, the rest of the residents benefit by not doing so.

Below is a general payoff matrix for the Volunteer's Dilemma:

https://en.wikipedia.org/wiki/Volunteer%27s_dilemma

In the case of the electricity in Jencks, if you decide to volunteer you're payout will be 0 no matter what because of the cost you endure when you decide to volunteer.

However, if you do not decide to volunteer, you have to bank on someone else volunteering.  In the case that you do not call, but someone else does, you're payoff is 1 because the electricity is back and you did not endure the cost of calling.  But if you decide not to volunteer, and no one else does we have the worst possible payout of -10 where the electricity is not returned to Jencks.

When the Volunteer's Dilemma is played between only two players, we have a situation like the game of "chicken" that Nevin introduced in his presentation where one person needs to "volunteer" to swerve out of the way, or we have the worst possible outcome. 

Now that we have a basic understanding of the Volunteer's Dilemma, do you guys think that a dominant strategy exists?  Feel free to post your thoughts in the comments.

The Murder of Kitty Genovese

Kitty Genovese was a bartender who was on her way home from work late one night.  As she approached her apartment building, she was stabbed by someone who then fled the scene.  Even though it was late, several of Kitty's neighbors saw the incident from the windows of their apartment rooms.  Yet, no one called the police and Kitty bled to death.

We can think of "The Murder of Kitty Genovese" as a two-player game that involves the Volunteer's Dilemma.

Let's say you and your neighbor are the only two witnesses who saw Kitty get stabbed.  You both know that if one person calls the police, an ambulance will arrive quickly and Kitty's life will be saved.  But, you cannot observe whether your neighbor has called and vice versa.  Both of you value her survival at 1 and her death at 0.  Calling the police has a cost of c where 0 < c < 1.  Hence, we get a payoff matrix that looks like this:

https://www.youtube.com/watch?v=FJf5Iw9dDzk

Now, let's consider the ideal mixed strategy for Player 1.  This means we need to find out what probability Player 1 should volunteer and what probability Player 1 should not volunteer.  

Let's start by finding the expected value of not volunteering.  If Player 1 does not volunteer we know that the payoff will either be 0 or 1.  We can solve for the expected value of not volunteering by using the equation:

EVI = PI(0) + (1 - PI)(1)

Where EVI is the expected value for Player 1 of ignoring and PI is the probability that Player 2 will ignore.

The expected value of calling is a bit less interesting since it does not rely on Player 2's decision and always equals 1 - c. So, we get:

EVC = 1 - c

Where EVC is the expected value of calling for Player 1.

When we set these equations equal to each other we get: PI(0) + (1 - PI)(1) = 1 - c, which simplifies to PI = c.  This means that the probability that Player 2 will ignore should be c and the probability Player 2 will call is 1 - c.  Since this game is symmetric, the same probabilities apply to Player 1's mixed strategy.

Since 0 < c < 1, this means that there is a positive probability that both players do not call the police and let Kitty die, even though they would both rather her live than die.  Additionally, this means that as the cost of calling the police increases, so does the probability that no one will call.

In a two player game, the probability that nobody volunteers is c^2.  This means that as we add more players to the game, or more witnesses to Kitty's murder there is more of a chance that no one will call.  In psychology, this is known as the Bystander Effect.

Please feel free to post any questions that you guys have, and I will try my best to cover them during my presentation.

Sources:

Grossman, Molly, Mandy Korpusik, and Philip Loh. "Chapter 14  Case Study: The Volunteer’s Dilemma." Case Study: The Volunteer’s Dilemma. N.p., n.d. Web. 09 Apr. 2017. <http://greenteapress.com/complexity/html/thinkcomplexity015.html>.

Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books. ISBN 0-385-41580-X.

Weesie, Jeroen (1993). "Asymmetry and Timing in the Volunteer's Dilemma". Journal of Conflict Resolution. 37 (3): 569–590. doi:10.1177/0022002793037003008. JSTOR 174269

Game Theory 101: The Murder of Kitty Genovese (Volunteer's Dilemma). Perf. William Spaniel. YouTube. Web. <https://www.youtube.com/watch?v=FJf5Iw9dDzk>.