The Volokh Conspiracy -

Rational Voting: Second Lindgren Post (Responding to Somin).--

I can see that Ilya Somin and I will probably go a couple more rounds in our discussion of rational voting.

Nonetheless, I confess that I was surprised by Ilya's response in a post below.

Here is Ilya's first point:

I assumed, in my analysis, that they value benefits to fellow citizens on average, 1/1000 as much as they value benefits to themselves. Jim argues, however, that:

Ilya's equation assumes that, if a voter could guarantee a victory for his preferred candidate, a typical voter would be willing to pay only $5,000 for one person's benefit (presumably his own), but that the same voter would be willing to pay about $1.5 billion dollars to benefit others ($5,000 x 300 million people / 1000). In other words, Ilya assumes that a rational voter when voting values the total utility of other Americans 300,000 times more than he values his own total non-altruistic utility ($1.5 billion to $5,000). Moreover, even leaving aside the comparative valuation, it can't be that (because of altruism) the utility to each person voting of having one's preferred candidate certain to win would be $1.5 billion dollars. To say that these are extraordinarily implausible assumptions is an understatement.
I have two responses to Jim's point, one technical, the other intuitive. Let's take the intuitive point first: Jim's analysis assumes that the relationship between the amount of money you are willing to give up to benefit others and the amount of benefit they receive from the sacrifice is purely linear. That is, if you are willing to give up $1 so that your neighbor will get $1000, you are also willing to give up $1.5 billion in order to give your fellow Americans $1.5 trillion. To my mind, the second doesn't necessarily follow from the first. Jim has shown that my analysis becomes implausible in cases where the voter/citizen is called upon to make very large sacrifices. When we're talking about voting, we're generally talking about a very small sacrifice.

But Ilya, surely you understand that it is YOUR EQUATION that is linear. If you run the numbers through your own equation, you get EXACTLY the results I reported. You assume that each and every voter on average would value a guarantee of his own preference for president at $5,000 for one person's benefit and altruistically internalizes 1.5 billion dollars of the benefit of others.

Dividing the $1.5 billion altruistic utility that your equation and example say that each voter on average acts on in choosing to vote by 100 million to generate a benefit of $15 does not make your equation any less linear, a fact that you surely know. If the relationship is not "purely linear" (or even remotely linear), then your equation is dead wrong because your equation is purely linear.

Really, you can forget about whether the supposed $1.5 billion converts to income in a linear fashion, since my critique does not actually depend on whether you are now partially undercutting your earlier claim that the utility you posit has direct dollar equivalents.

As my own speculative example hinted at, I doubt that most voters valuing a win for their preferred candidate at $5,000 (your assumed utility value) would altruistically value and internalize into their own decisionmaking the utility of victory for others at more than $5,000-50,000, certainly nothing even close to the $1.5 billion value for altruism you posit as being present on average.

Remember, you need such a huge altruistic value for total utility or your hypothesized rationality doesn't appear. Would someone rationally incur a $10 loss to gain a 0.005 cent to 0.05 cent gain in utility, as my more reasonable assumption of total utility would generate? Even if your model had assumed nonlinearity or nonequivalence in money (neither of which you actually assumed in your article), there still would be a massive gap between the 1.5 billion in total internalized utility that you assume on average and the $5,000 to $50,000 total internalized utility that I would assume on average.

I guess I've persuaded you that your equation is wrong, because in your response you appear not to think that each and every voter on average values the winner at 1.5 billion dollars, which is the number that your equation and your hypothetical example yields.

(Click to show the rest of the post)

In generating the $1.5 billion estimate, I didn't use a different equation or even different numbers than you use in your example on page 4 of your manuscript: you value the expected total utility of getting one's preference for office for each voter at 1.5 billion dollars per voter on average.

I can only conclude that you find $1.5 billion dollars of altruistic commitment per voter (and your equation and your example that generated it) as faulty as I do.

Ilya's Second Argument (related to the first):

Second, the technical point. Jim has (understandably) conflated the distinction between dollar income and utility. My analysis assumes that people value the utility of others at 1/1000 of their own, which is not the same thing as valuing the added dollar income of others at 1/1000 of the rate at which you value adding dollars to your income. When we're talking about making a sacrifice of $35,000 out of a $50,000 annual income (to use Jim's example), we're talking about a vastly greater loss of utility to the donor than when we're talking about sacrificing $10. And the difference between the two may well be much greater than $35,000/10. The $10 sacrifice is essentially trivial, while the $35,000 may wreck the donor's life for months or years to to come.

Ilya, I can't believe that you would accuse me of conflating dollar income with utility given what you write in your article (whose equations you told me to look at). You wrote:

Assume that Uv = expected utility of voting; Cv = cost of voting; and D=expected difference in welfare per person if the voter's preferred candidate defeats her opponent. Let us further assume that this is a presidential election in a nation with 300 million people; that the voter's ballot has only a 1 in 100 million chance of being decisive (Riker and Ordeshook 1968); and that the voter values the welfare of his fellow citizens an average of 1000 times less than his own. Thus, we get the following equation:

Equation 1: The Utility of Voting
D*(300 million/1000)/(100 million) -- Cv =Uv.

If we assume that Cv is $10 (a reasonable proxy for the cost of voting) and that D is $5000 (this can incorporate monetary equivalents of noneconomic benefits as well as actual income increases), then Uv equals $5, a small but real positive expected utility.

OK, so that's exactly what I assumed in my analysis: that D (the "expected difference in welfare per person if the voter's preferred candidate defeats her opponent") was $5,000, that D "can incorporate monetary equivalents of noneconomic benefits as well as actual income increases," that (D times (300 million/1000)), which is 1.5 billion dollars for each voter on average, is the total expected utility per voter of victory expressed in dollars in your example. All this is exactly the same for both of us.

So after you convert utility to dollars, and after you specifically say that these represent "monetary equivalents of noneconomic benefits," you now complain that I take you seriously that these are actually monetary equivalents.

Further, as you must know, my analysis does not depend on any monetary transfers or income being made. Your example and equation claims that each and every voter on average makes decisions to vote on his internalization of a 1.5 billion dollar total utility for determining the outcome. I point out how unrealistic that assumption is by asking whether most voters would trade that $1.5 billion utility for more than a billion dollars in cash. Surely you must realize that my questioning of your $1.5 billion total internalized utility for each voter doesn't depend on any money actually changing hands, since a near guarantee of victory would almost never be possible. The $1.5 billion total is completely implausible on its face, especially given that you assume that the non-altruistic utility of any one voter is only $5,000 on average.

If you are unwilling to defend the $1.5 billion total utility generated by your example and equation as approximately correct, then you shouldn't be defending that same number when it is divided by 100 million, since the resulting estimate of $15 is derived from a total altruistic utility per voter that no one could possibly find plausible.

Similarly, my example of a voter with a family gross income of $50,000 and a take-home pay of $35,000 was simply to illustrate that an extremely altruistic person would never internalize $1.5 billion of benefits to others. Go ahead and less realistically assume that the average voter makes instead $200,000 or $10,000. That makes virtually no difference whether it is plausible to assume that such a typical voter would value other people's interest in getting the right president at $1.5 billion, which would equal many times the voter's income for the rest of his or her life, whether any transfer would ever take place.

I really don't think that you should convert utility to dollar equivalents, including "monetary equivalents of noneconomic benefits," and then criticize me for "conflat[ing] the distinction between dollar income and utility." And I don't think that you should use a purely linear equation and a purely linear example and then complain when I point out the implausibility of the totals that such a linear equation generates.

Are you sure, Ilya, that you don't want to rethink this paper? It's no shame to withdraw a piece for a complete rethinking after it's accepted for publication because basic problems appear after acceptance. I've done it.

OK, Ilya, your turn.

(hide)

	[Jim Lindgren, November 6, 2006 at 1:35am] Trackbacks Rational Voting: Second Lindgren Post (Responding to Somin).-- I can see that Ilya Somin and I will probably go a couple more rounds in our discussion of rational voting. Nonetheless, I confess that I was surprised by Ilya's response in a post below. Here is Ilya's first point: I assumed, in my analysis, that they value benefits to fellow citizens on average, 1/1000 as much as they value benefits to themselves. Jim argues, however, that: Ilya's equation assumes that, if a voter could guarantee a victory for his preferred candidate, a typical voter would be willing to pay only $5,000 for one person's benefit (presumably his own), but that the same voter would be willing to pay about $1.5 billion dollars to benefit others ($5,000 x 300 million people / 1000). In other words, Ilya assumes that a rational voter when voting values the total utility of other Americans 300,000 times more than he values his own total non-altruistic utility ($1.5 billion to $5,000). Moreover, even leaving aside the comparative valuation, it can't be that (because of altruism) the utility to each person voting of having one's preferred candidate certain to win would be $1.5 billion dollars. To say that these are extraordinarily implausible assumptions is an understatement. I have two responses to Jim's point, one technical, the other intuitive. Let's take the intuitive point first: Jim's analysis assumes that the relationship between the amount of money you are willing to give up to benefit others and the amount of benefit they receive from the sacrifice is purely linear. That is, if you are willing to give up $1 so that your neighbor will get $1000, you are also willing to give up $1.5 billion in order to give your fellow Americans $1.5 trillion. To my mind, the second doesn't necessarily follow from the first. Jim has shown that my analysis becomes implausible in cases where the voter/citizen is called upon to make very large sacrifices. When we're talking about voting, we're generally talking about a very small sacrifice. But Ilya, surely you understand that it is YOUR EQUATION that is linear. If you run the numbers through your own equation, you get EXACTLY the results I reported. You assume that each and every voter on average would value a guarantee of his own preference for president at $5,000 for one person's benefit and altruistically internalizes 1.5 billion dollars of the benefit of others. Dividing the $1.5 billion altruistic utility that your equation and example say that each voter on average acts on in choosing to vote by 100 million to generate a benefit of $15 does not make your equation any less linear, a fact that you surely know. If the relationship is not "purely linear" (or even remotely linear), then your equation is dead wrong because your equation is purely linear. Really, you can forget about whether the supposed $1.5 billion converts to income in a linear fashion, since my critique does not actually depend on whether you are now partially undercutting your earlier claim that the utility you posit has direct dollar equivalents. As my own speculative example hinted at, I doubt that most voters valuing a win for their preferred candidate at $5,000 (your assumed utility value) would altruistically value and internalize into their own decisionmaking the utility of victory for others at more than $5,000-50,000, certainly nothing even close to the $1.5 billion value for altruism you posit as being present on average. Remember, you need such a huge altruistic value for total utility or your hypothesized rationality doesn't appear. Would someone rationally incur a $10 loss to gain a 0.005 cent to 0.05 cent gain in utility, as my more reasonable assumption of total utility would generate? Even if your model had assumed nonlinearity or nonequivalence in money (neither of which you actually assumed in your article), there still would be a massive gap between the 1.5 billion in total internalized utility that you assume on average and the $5,000 to $50,000 total internalized utility that I would assume on average. I guess I've persuaded you that your equation is wrong, because in your response you appear not to think that each and every voter on average values the winner at 1.5 billion dollars, which is the number that your equation and your hypothetical example yields. (Click to show the rest of the post) In generating the $1.5 billion estimate, I didn't use a different equation or even different numbers than you use in your example on page 4 of your manuscript: you value the expected total utility of getting one's preference for office for each voter at 1.5 billion dollars per voter on average. I can only conclude that you find $1.5 billion dollars of altruistic commitment per voter (and your equation and your example that generated it) as faulty as I do. Ilya's Second Argument (related to the first): Second, the technical point. Jim has (understandably) conflated the distinction between dollar income and utility. My analysis assumes that people value the utility of others at 1/1000 of their own, which is not the same thing as valuing the added dollar income of others at 1/1000 of the rate at which you value adding dollars to your income. When we're talking about making a sacrifice of $35,000 out of a $50,000 annual income (to use Jim's example), we're talking about a vastly greater loss of utility to the donor than when we're talking about sacrificing $10. And the difference between the two may well be much greater than $35,000/10. The $10 sacrifice is essentially trivial, while the $35,000 may wreck the donor's life for months or years to to come. Ilya, I can't believe that you would accuse me of conflating dollar income with utility given what you write in your article (whose equations you told me to look at). You wrote: Assume that Uv = expected utility of voting; Cv = cost of voting; and D=expected difference in welfare per person if the voter's preferred candidate defeats her opponent. Let us further assume that this is a presidential election in a nation with 300 million people; that the voter's ballot has only a 1 in 100 million chance of being decisive (Riker and Ordeshook 1968); and that the voter values the welfare of his fellow citizens an average of 1000 times less than his own. Thus, we get the following equation: Equation 1: The Utility of Voting D*(300 million/1000)/(100 million) -- Cv =Uv. If we assume that Cv is $10 (a reasonable proxy for the cost of voting) and that D is $5000 (this can incorporate monetary equivalents of noneconomic benefits as well as actual income increases), then Uv equals $5, a small but real positive expected utility. OK, so that's exactly what I assumed in my analysis: that D (the "expected difference in welfare per person if the voter's preferred candidate defeats her opponent") was $5,000, that D "can incorporate monetary equivalents of noneconomic benefits as well as actual income increases," that (D times (300 million/1000)), which is 1.5 billion dollars for each voter on average, is the total expected utility per voter of victory expressed in dollars in your example. All this is exactly the same for both of us. So after you convert utility to dollars, and after you specifically say that these represent "monetary equivalents of noneconomic benefits," you now complain that I take you seriously that these are actually monetary equivalents. Further, as you must know, my analysis does not depend on any monetary transfers or income being made. Your example and equation claims that each and every voter on average makes decisions to vote on his internalization of a 1.5 billion dollar total utility for determining the outcome. I point out how unrealistic that assumption is by asking whether most voters would trade that $1.5 billion utility for more than a billion dollars in cash. Surely you must realize that my questioning of your $1.5 billion total internalized utility for each voter doesn't depend on any money actually changing hands, since a near guarantee of victory would almost never be possible. The $1.5 billion total is completely implausible on its face, especially given that you assume that the non-altruistic utility of any one voter is only $5,000 on average. If you are unwilling to defend the $1.5 billion total utility generated by your example and equation as approximately correct, then you shouldn't be defending that same number when it is divided by 100 million, since the resulting estimate of $15 is derived from a total altruistic utility per voter that no one could possibly find plausible. Similarly, my example of a voter with a family gross income of $50,000 and a take-home pay of $35,000 was simply to illustrate that an extremely altruistic person would never internalize $1.5 billion of benefits to others. Go ahead and less realistically assume that the average voter makes instead $200,000 or $10,000. That makes virtually no difference whether it is plausible to assume that such a typical voter would value other people's interest in getting the right president at $1.5 billion, which would equal many times the voter's income for the rest of his or her life, whether any transfer would ever take place. I really don't think that you should convert utility to dollar equivalents, including "monetary equivalents of noneconomic benefits," and then criticize me for "conflat[ing] the distinction between dollar income and utility." And I don't think that you should use a purely linear equation and a purely linear example and then complain when I point out the implausibility of the totals that such a linear equation generates. Are you sure, Ilya, that you don't want to rethink this paper? It's no shame to withdraw a piece for a complete rethinking after it's accepted for publication because basic problems appear after acceptance. I've done it. OK, Ilya, your turn. (hide) Related Posts (on one page): Still More on Rational Voting - 2nd Reply to Jim Lindgren: Rational Voting: Second Lindgren Post (Responding to Somin).-- Comments [Ilya Somin, November 6, 2006 at 2:27am] Trackbacks Still More on Rational Voting - 2nd Reply to Jim Lindgren: Jim Lindgren's most recent post is a rejoinder to my original reply. To recap, I argued that Jim's initial argument against my theory of rational voting doesn't refute it because Jim assumes that, for my theory to be correct, there must be a linear relationship between "the amount of money you are willing to give up to benefit others and the amount of benefit they receive from the sacrifice... That is, if you are willing to give up $1 so that your neighbor will get $1000, you are also willing to give up $1.5 billion in order to give your fellow Americans $1.5 trillion." To my mind, there is a big difference between the willingness to make relatively small, even trivial sacrifices and very large ones. The person who is willing to sacrifice $10 to give his fellow citizens an expected benefit of $15,000 (the tradeoff in the equation from my paper) won't necessarily be willing to sacrifice $1.5 million to give them $1.5 trillion. Jim replies: But Ilya, surely you understand that it is YOUR EQUATION that is linear. If you run the numbers through your own equation, you get EXACTLY the results I reported. You assume that each and every voter on average would value a guarantee of his own preference for president at $5,000 for one person's benefit and altruistically internalizes 1.5 billion dollars of the benefit of others. Dividing the $1.5 billion altruistic utility that your equation and example say that each voter on average acts on in choosing to vote by 100 million to generate a benefit of $15 does not make your equation any less linear, a fact that you surely know. If the relationship is not "purely linear" (or even remotely linear), then your equation is dead wrong because your equation is purely linear. Yes, the equation is purely linear. That is because I did not consider, in setting up the model, cases where the decisionmaker must decide whether to make a very large sacrifice rather than a small one. Quite likely, I was wrong to ignore that scenario, although I don't think that it is relevant to an analysis of the utility of voting (almost always an act that requires only small sacrifices). But building it into the equation would not significantly alter the result. For example, one could change the cost term in the equation (cost of voting) to include an additional variable that accounts for the fact that the tradeoff will change as C (the dollar cost of voting) becomes an increasingly high percentage of the potential voter's total wealth. Thus, the Total Cost (TC) of voting=C + (CxP), with P representing the percentage of the voter's wealth that $C is. For most people, however the cost of voting (which I assumed to be $10) will be a tiny percentage of their total wealth, and thus CxP will be utterly insignificant in real world terms (at least when dealing with voting). For example, if $10 is an unrealistically high 0.1% of the person's total wealth, then CxP will be $1. One could even posit a geometric rather than arithmetic increase in the value of CxP as C rises, since sacrificing 50% of your wealth may well pinch a lot more than fifty times as much as the suffering you feel from sacrificing %1.In technical economic terms, the marginal utility of income may go down as your wealth increases, but it may actually go up as your wealth decreases. Losing $100 is a small sacrifice for me, but perhaps a big one for a very poor person. An implication of this point is that very poor people will have lower turnout rates than others (because they are the only ones for whom CxP might be significant), and the evidence of numerous turnout studies does indeed bear this out. In sum, the unwillingness of people to make very large sacrifices to vote, in my view, says little about their willingness to make very small ones. However, Jim's analysis does highlight the need to deal with this point explicitly in future versions of the model (sadly, it is too late to incorporate in this particular publication, which will soon be going to press). Lastly, Jim, I think, continues to make the mistake of conflating dollar cost with utility, that I pointed out in my previous post. In the paper, I use hypothetical dollars as a convenient symbol for total utility, but I also emphasized that, in the real world, monetary cost is just one part of the total utility of voting. UPDATE: Here's a simpler way of making my point that occurred to me after I'd already written the body of this post: Many people routinely contribute small portions of their income to charity, say $100, in order to produce benefits for others. Let's say that a $100 donation produces 100 units of utility for those who benefit from the charity. Most of those who are willing to donate $100 to charity in order to produce 100 utiles would not be willing to donate $50,000 to produce 50,000 utiles. Yet this is not irrational; it simply represents the fact there is a nonlinear relationship between people's willingness to make small sacrifices for others and their willingness to make large ones. Many people are willing to be moderately generous, but very few are willing to be martyrs for a cause. The same point applies to voting. The person who is willing to sacrifice the equivalent of $10 in order to give his fellow citizens an expected value of $15,000 is not necessarily irrational if she is simultaneously unwilling to sacrifice $1.5 million in order to give them $1.5 trillion, or even to sacrifice $35,000 in order to do so (as in one of the examples Jim analyzes). Related Posts (on one page): Still More on Rational Voting - 2nd Reply to Jim Lindgren: Rational Voting: Second Lindgren Post (Responding to Somin).-- Comments