Flipping Coins and Saving Lives
September 4, 2009 — 10:14

Author: Michael Almeida  Category: Uncategorized  Comments: Off

[Modified and Updated 9-7]
Suppose you find yourself in a situation where you can save the lives of each of three persons A, B, or C, but cannot save all. Each is equally worthy of being saved, none is responsible for the life-threatening situation, etc. The *Equal Greatest Chance* principle (EGC) states that, when all else is equal, we should act in ways that give each person an equal greatest chance of surviving. If I have to choose between saving A and saving B, for instance, and all else is equal, then I should flip a fair coin or otherwise give each a .5 chance at surviving. But now suppose I have to choose between saving A or saving both B & C. Perhaps A is on one island and about to die, and B & C are on another and about to die. I cannot save everyone. In order to give each person an equal chance of surviving, it seems, I have to flip a fair coin. In any case, this is the conclusion of Jim Taurek (‘Should Numbers Count?’ *PPA*, 1977).
Ben Bradley presents what he claims is a decisive counterexample to (EGC) (see *Journal fo Ethics and Social Philosophy*, March, 2009). Against (EGC) Bradley puts the Save the Greatest Number principle (SGN). In cases that involve saving lives such as the one noted above, we should act in ways that save the greatest number of persons. But what is Bradley’s counterexample to (EGC)? It begins this way,
> Here is the example. Batman believes in EGC, and never leaves home without
> a fair coin. Batman’s nemesis, a murderous bureaucrat named the Joker,
> has captured three hostages, named Alice, Bob and Carol. The Joker tells
> Batman the following: “I am going to divide these three hostages randomly
> into two groups – a group of two and a group of one. I will let you determine
> which group you wish to save, and I will kill only the members of the other
> group. Indicate your decision by filling out this form, and checking the appropriate
> box.” As a believer in EGC, Batman would choose to save the
> larger group, since that decision gives each a two-thirds chance of survival,
> and nothing else gives each person a greater chance. If Batman were to flip a
> coin to decide whether to save the larger or the smaller group, he would diminish
> each person’s chance of survival to one-half. So far, EGC and SGN
> get the same results.

But how does Bradley arrive at the conclusion that Batman must save the greater number? The reasoning seems to go this way,
1. The Joker randomly divided A, B and C into two groups X and Y.
2. If the Joker randomly divided A, B, and C into two groups X and Y, then the chances that A is in the larger group (similarly for B and C) is two thirds.
3. Therefore, I give each person a 2/3 chance of surviving if I choose the larger group.
But that’s not valid. What follows is not (3), but (3′).
3′. Therefore each person has a 2/3 chance of being in the larger group.
Why doesn’t (3) follow from (1) and (2)? Suppose (1)-(2) are true and I see that A is in the smaller group and B and C are in the larger group. If I then choose the larger group, I do not give A a 2/3 chance of surviving. In fact, I give A no chance of surviving at all. This is consistent with it being true that, when the story of A is told, someone says that she had a 2/3 chance of surviving this whole incident, but died.
Suppose then that I did not see who was in the larger group and who was in the smaller group. If I then choose the larger group, then I do give A a 2/3 chance of surviving.
So, the counterexample to (EGC) is going to fail, since this part of the argument–the argument from (1) & (2) to (3)–is invalid. We should come to these two unusual conclusions about such cases.
C1. If I have the information about who is in the larger and who is in the smaller group, then I should flip a coin to give each person Equal Greatest Chance at surviving.
C2. If I do not have the information about who is in the larger and who is in the smaller group, then I should choose the larger group to give each person the Equal Greatest Chance at surviving.
Final twist. Suppose what I do is adopt a policy of always choosing the larger group. Now suppose I see that A is in the smaller group and B & C are in the larger group. In that case, if I stay with my policy, C3 is true.
C3. A had a 2/3 chance of surviving, but I gave A no chance of surviving.
So the central question is whether someone committed to (EGC) should observe who is in each group, or not observe who is in each group. The former practice seems more like ostridge decision making, and it is just bizarre to think it gives each person a greater chance of surviving. The latter, on the other hand, certainly cuts against intuitions that we should save the greater number.