Thomas R. Palfrey and Jeffrey E. Prisbrey in "Anomalous Behavior in Public Goods Experiments: How Much and Why?" American Economic Review, Volume 87, Issue 5 (Dec., 1997), pp. 829-846, examine the individual player contribution rates as a function of the investment costs by assigning different rates of return from the private consumption in a voluntary contribution experiments and directly test these response functions for the presence of warm-glow and altruism effects.

Each participant of the public good's game was given an endowment which could be invested in a public good or kept for private consumption as in a standard voluntary contribution mechanism (VCM) setting. All the subjects in the group had the same marginal valuation of the public good, known to everybody. But the individual subjects were assigned different marginal values for the private good. This way, some subjects had a dominant strategy to contribute all of their endowment to the public good and some had a dominant strategy to keep it and free ride and these incentives changed each period. The monetary payoff function for each individual was of the form:

U(x_i, x_-i) = V _j x_j + r_i(w_i - x_i),

where x_i is the individual's contribution, V is the marginal value of the public good and is the same for each subject, and r_i is the marginal value of the private good known only to that individual.

In previous experimental studies the marginal valuations of the private good were the same for all subjects in all periods and usually the private good valuations exceeded the valuation of public good, thus, creating an incentive to free ride. The main findings are reported below:

• "Most players in this game violate their one-shot dominant strategy, with many contributing upwards of half their endowment. They do so even when the marginal valuation is three or more times that of the public good.
• As the marginal valuation of the private good gets closer to the marginal valuation of the public good, more violations of the dominant strategy are observed.
• Subjects can be roughly categorized according to their tendency to violate the dominant strategy.
• Violations of dominant strategies diminish both with repetition and with experience (playing a second sequence of games with a new group).
• Violations of dominant strategies to contribute appear to be as prevalent as violations of dominant strategies to free ride." (Palfrey and Prisbrey [1997], pp. 830).

• altruistic preferences (increasing utility not only in one's own payoff but also in the group payoff)
• warm-glow preferences (the act of contributing, independent of how much it increases the group payoffs, increases one's utility by a fixed amount)
• repeated game effects, such as reputation building and learning
• subject confusion

Each subject participated in four rounds of ten periods where each ten-periods round took place with a different group of three other subjects. The first two and the last two rounds had the same marginal valuation of the public good, but these were different from one another. All four rounds occured in one session and each session included 16 subjects.

The subjects received private good values randomly assigned according to a known uniform distribution between 1 and 20 in unit increments each time when making a new decision. This way the data contained multiple observations of the choices of each subject at different marginal values of private good. Such design allowed to estimate the response functions at both individual and aggregate levels.

The value of the public good varied between experiments (V was equal to 3, 6, 10, and 15) and so did the endowment (equal either 1 or 9 tokens).

In the data analysis part the authors focus on two issues, namely, identify errors or background noise (behavior inconsistent with standard theory) and attempt to measure response functions, i.e., dependence of contribution decisions on the private token values and the public good value. They econometrically analyze the data and test for the presence of two components - the altruism and warm-glow effects. The altruism effect informs about the additional utility gained from increasing the monetary payoff to other participants in the group by one unit. The warm-glow effect, on the other hand, informs about the additional utility gained from the act of contributing a unit of endowment towards a public good. The way the presence of the two effects can be detected is as follows. Altruism is present in subjects' behavior if contributions increase as the value of the public good increases, ceteris paribus. Warm-glow effect can be found if contributions towards the public good increase as the difference between the public good value and the token value increases, ceteris paribus. Since both the token values and public good values vary in the experimental design, these two effects, if present, can easily be indentified.

Since the estimation of probability that a particular subject will contribute towards a public good given the own value for the private good and the common value of the public good requires advanced econometric methods what is beyond the scope of this handbook, we omit these results and rather report the findings on the warm-glow, altruism, and subjects' errors. The advanced reader interested in learning about the details of probability estimation should consult the original article itself.

Palfrey and Prisbrey found that "... altruism played little or no role at all in the individual's decision and, on the other hand, warm-glow effects and random error played both important and significant roles" (pp. 842). Furthermore, the experimental data show that experience is a significant determinant of contributions and as the experience grows, the contribution rates decline. The decline is mainly a result of a reduction in subject decision error and a lower variance in distribution of individual warm-glow effects, but not an overall decrease of warm-glow level. This suggests that the subjects do not become more selfish with experience but their preferences are rather stable. The data consistently replicate common observations from other public goods experiments: decay, experience, and the strong responsiveness of contribution rates to the opportunity cost of contribution.