In his piece "Does Vegetarianism Make a Difference?", Alan Dawrst argues that refraining from eating a given number n of animals may not save n animals, but has an effect that is ethically equivalent: it diminishes by that same amount the expected value of the number of animals who will be raised and slaughtered.

I think the issue is of some importance, and I agree with Alan's conclusion. I also agree with his reasoning:

The problem with this reasoning is that it is dependent on the specifics of particular situations: crates of wings, sheds of chickens and so on. It is easy for each example to appear to be just that: a particular example, disconnected from other cases.

There is a simpler and more general way to get to the same conclusion.

The basic idea is that probabilities, and hence expected values, are essentially subjective, that is depend on what we do or do not know. Now, as Alan says, "[e]veryone in the debate agrees that, at some point, a substantial decrease in demand for meat--for instance, conversion to vegetarianism by half of the US population---would diminish the quantity supplied of factory-farmed animals". Given what we know, we can plot the expected value N(n) of the number of animals who will be raised and slaughtered against the number n of animal bodies that are eaten. That curve will be such that, for instance, if n was halved, the expected value N(n) would be halved too: N(n/2) = N(n) / 2. Actually, for any set of far-spaced values of n, such as in figure 1, the expected value will be approximately proportional to n:

Between those points, how will the curve behave? It is true that the real number of animals raised and slaughtered will vary by large chunks, such as when an investor decides whether to build another shed for producing batches of a hundred thousand broiler chickens. So we might expect the curve to be as in figure 2:

But the fact is that, barring particular circumstances, we have no idea at what points such discontinuities will occur. Since the expected value is a function of what we know, it cannot reflect any such discontinuities. It will have to be smooth:

This means that any variation, however small, in the value of n will produce a correspondingly small variation in N(n). Eating one less chicken will be ethically equivalent to saving one chicken.

The difficulty people have in grasping this is that they tend to switch back and forth between the expected value and the real value.

This line of reasoning is more formal and general than Alan's. I think that it does not actually replace it; rather, it is complementary. Alan's reasoning is more concrete, because it shows the particular way things can be in specific cases.

I think the issue is of some importance, and I agree with Alan's conclusion. I also agree with his reasoning:

Assume it takes 200 fewer consumers of chicken wings in order for the supermarket to buy one less case. Inasmuch as individual consumers have no way of telling whether their particular wing will be the one that changes the number of cases purchased, the probability of any given carton being the determining factor is 1/200. The expected value of an action is the probability that a benefit will result times the magnitude of the benefit if it does result, so the expected value of refraining from the purchase of any given carton of milk is (1/200)(1 fewer case purchased)(200 wings/case) = 1 fewer wing purchased. The exact expected values will of course fluctuate on account of the randomness of the purchasing agent's decisions (if, for instance, she would not buy one fewer case until 300 fewer consumers demanded wings, even though each case includes only 200 wings), but they should average out over the long run in such a way that forbearing the purchase of any given amount of an animal product will be expected to reduce bulk purchase of that amount of the product.

This logic applies also to the rest of the factory-farmed-meat demand process: at some critical mass of fewer cases ordered by stores, distributors will purchase fewer chicken wings from farms, and that reduced demand from farms will, at some point, constrict production. By the end, the probability that any given consumer will impact animal production is miniscule, but the benefits if he does are immense. Thus, the expected value of refraining from the purchase of any given amount of an animal product is roughly equivalent to preventing the production of the portion of an animal that the product represents.

The problem with this reasoning is that it is dependent on the specifics of particular situations: crates of wings, sheds of chickens and so on. It is easy for each example to appear to be just that: a particular example, disconnected from other cases.

There is a simpler and more general way to get to the same conclusion.

The basic idea is that probabilities, and hence expected values, are essentially subjective, that is depend on what we do or do not know. Now, as Alan says, "[e]veryone in the debate agrees that, at some point, a substantial decrease in demand for meat--for instance, conversion to vegetarianism by half of the US population---would diminish the quantity supplied of factory-farmed animals". Given what we know, we can plot the expected value N(n) of the number of animals who will be raised and slaughtered against the number n of animal bodies that are eaten. That curve will be such that, for instance, if n was halved, the expected value N(n) would be halved too: N(n/2) = N(n) / 2. Actually, for any set of far-spaced values of n, such as in figure 1, the expected value will be approximately proportional to n:

Between those points, how will the curve behave? It is true that the real number of animals raised and slaughtered will vary by large chunks, such as when an investor decides whether to build another shed for producing batches of a hundred thousand broiler chickens. So we might expect the curve to be as in figure 2:

But the fact is that, barring particular circumstances, we have no idea at what points such discontinuities will occur. Since the expected value is a function of what we know, it cannot reflect any such discontinuities. It will have to be smooth:

This means that any variation, however small, in the value of n will produce a correspondingly small variation in N(n). Eating one less chicken will be ethically equivalent to saving one chicken.

The difficulty people have in grasping this is that they tend to switch back and forth between the expected value and the real value.

This line of reasoning is more formal and general than Alan's. I think that it does not actually replace it; rather, it is complementary. Alan's reasoning is more concrete, because it shows the particular way things can be in specific cases.