How Donor Uncertainty Affects Their Response to Matches

The author gratefully acknowledges support from the National Science Foundation (Award Number: 2315706) and the Department of Agricultural and Resource Economics (Bessie H. DeVault Award and The Bruce and Mary Ann Gardner Dissertation Enhancement Award).

Lead gifts are often used by charities to match donors' contributions in an attempt to increase charitable receipts. However, previous research has found inconsistent effects on giving. In some cases, increasing the match rate, \(m\), is observed to increase charitable receipts, while in other cases it is observed to decrease charitable receipts. This leaves charitable organizations with little guidance for determining the optimal match rate to offer, and it provides a puzzle for theoretical models of giving.

In this paper I suggest a possible explanation for the variability in responses to matches observed in previous studies. Namely, I argue that a donor's response to a match may be affected by their beliefs regarding the probability with which their donation will actually be matched at the margin. The intuition is straightforward: if total giving is high enough to fully use up the lead gift, any additional donations will not be matched. Thus, for a given lead gift, \(\phi\), increasing the match rate, \(m\), provides two counteracting effects for donor \(i\): it increases the potential impact of any donations provided by donor \(i\), but it also increases the probability that others' total donations will exceed the match limit (i.e., \(P\left( G_{-i} > \frac{\phi}{m}\right) \)), in which case donor \(i\)'s donation will not be matched at the margin.¹ Importantly, increasing the match rate (\(m\)) mechanically decreases the match limit (\(\frac{\phi}{m}\)), so that the donor's probability of receiving a match is directly affected by changes in \(m\), independent of any changes in their perceived distribution of \(G_{-i}\) resulting from the change in \(m\).

To formalize this explanation, I develop a theoretical model of giving in which donors view the total giving of others, \(G_{-i}\), as a random variable. Donor \(i\)'s beliefs about the distribution of \(G_{-i}\) are assumed to be a function of basic characteristics of the fundraiser, such as the the lead gift size (\(\phi\)) and the match rate (\(m\)). I make no assumptions regarding the structural form of subjects' beliefs about the distribution of \(G_{-i}\). Instead I show that, holding a donor's utility form constant, their response to a match (\(m\)) will still vary depending on their beliefs about how the probability of being matched (i.e., \(P\left( G_{-i} < \frac{\phi}{m}\right) \))² varies with respect to the characteristics of the fundraiser. Therefore, the model is able to explain the disparate responses to matches observed in previous studies using different fund-raising settings.

Finally, to test the performance of the model, I derive theoretical predictions from the model and design corresponding experiments to test those predictions. I first seek to demonstrate that donors are responsive to changes in the probability of being matched. I then seek to demonstrate that donors' beliefs about the probability of being matched are affected by changes in the characteristics of the fundraiser, and furthermore that these changes in their beliefs affect their response to matches in predictable ways.

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1. ¹ Technically, whether or not donor \(i\) receives a match on the margin also depends on their own decision of how much to give (i.e., \(g_i\)), so that the actual probability they are concernced with is \(P\left( g_i + G_{-i} > \frac{\phi}{m}\right) \) (or \(P\left( G_{-i} > \frac{\phi}{m} - g_i \right) \)). This accounts for the possibility that donor \(i\)'s donation happens to be the one that causes total donations to exceed the match limit. I am ignoring this possibility here because I feel that it distracts from the main intuition (and as a practical matter I believe this possibility can reasonably be assumed to be negligible in the vast majority of settings).
2. ² Again, technically the probability of being matched on the margin is actually given by \(P\left( G_{-i} < \frac{\phi}{m} - g_i \right) \). (See footnote 1.)

Lead gifts (which are large donations provided by wealthy donors) are often used by charities to aid fundraising efforts. One of the most common ways charities use lead gifts is to match other donors' donations. For example, after receiving a lead gift of \$1 million, a charity might inform potential donors that any donations they provide during the current fundraiser will be matched at a 1:1 rate by an anonymous donor up to a total amount of \$1 million (at which point the lead gift will be fully exhausted).

In general, providing a match to donors increases the total amount of donations received by the charity. However, previous studies have found conflicting results regarding the optimal match rate. That is, if a charity wants to maximize the total amount of donations it receives, should they provide a 1:1 match, a larger match (such as 2:1 or 3:1), or a smaller match (such as 0.75:1 or 0.5:1)? Some previous studies find that larger match rates lead to greater total donations, while others find that increasing the match rate actually decreases total donations, suggesting the use of smaller match rates (or no match at all). So what guidance should we provide to charities seeking to maximize their donations? And how can we make sense of the disparities between previous results?

In this paper I propose a simple explanation for why previous studies may find such disparate responses to matches: potential donors may care about the probability of actually receiving a match. Depending on the fundraiser and the size of the lead gift, increasing the match rate may lead donors to believe there is very little chance that the match limit will not be reached (in which case their donation will not actually be matched). Because of this, a donor who would otherwise respond positively to an increase in the match rate may instead respond negatively, if they believe the increase in the match rate will make them less likely to actually receive a match.

As an example, suppose that someone provides a lead gift of \$10,000 to your church and your church uses this money to offer a 1:1 match on donations. Suppose further that you believe there is little chance that total donations exceed \$3,000 dollars (so that only 30% of the lead gift will end up being used). In this case, if the church were to increase the match rate to 2:1 (so that each dollar you donate results in the charity receiving $3), you may be inclined to increase your donation, since you have a special opportunity to take advantage of the lead gift and increase the amount your church receives.

If, however, the $10,000 lead gift were instead provided to some large organization (e.g., the Red Cross) and used to offer a 1:1 match for some national fundraising campaign, you would likely be wholly unimpressed by the offer. Why? Because you would likely assume that total donations would greatly exceed the $10,000 limit, and your donation wouldn't actually be matched. And if the charity increased the match rate to 2:1, this would either have no effect on you or it may even cause you to decrease your donation. Why? Because now it only takes $5,000 in total donations for the match limit to be met, making it even more likely that you won't receive a match.

From the above example, we can see that an individual's response to a match will vary depending on the characteristics of the fundraiser in question and the individual's expectations regarding the total giving of others. Because increasing the match rate may sometimes lead donors to believe that they are less likely to actually be matched, donors may sometimes be observed to decrease their donations in response to increases in the match rate.

To formalize this argument, in the paper I build a theoretical model of giving that incorporates donors' beliefs about the probability of receiving a match, and I demonstrate that the model is capable of explaining previous results. I then derive testable predictions from the model and use economic lab experiments to test those predictions.