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# Discrete Public Goods

## Game Description

At issue is a discrete public good whose societal benefit is larger than its cost. It is only provided if voluntary contributions are at least as large as the cost (i.e., reach the threshold). This is a great environment to study free riding: there are multiple equilibria where the good is provided, but everyone prefers those equilibria in which they contribute less (or not at all).

Note: This is one of three MobLab public goods games. In Linear Public Goods and Public Goods (Signaling), the private returns from contributing are always less than the private returns from the alternative, but the latter game has the option of allowing players to punish or reward group members. Thus, while this game focuses on the coordination problem inherent in some voluntary contribution environments (everyone prefers those equilibria where her individual contribution is less), the Linear games highlight the distinction between private and societal benefits of a contribution.

### Learning Objective 1: Free-Rider Problem

Even if the societal benefit of a discrete public good exceeds its cost, relying on voluntary contributions may result in contributions insufficient to provide the good. The possibility that sufficient contributions will be collected (and therefore the good will be provided), even if the individual reduces or eliminates her contribution, provides an incentive to free ride.

### Learning Objective 2: Collective Action Problem

The free-rider problem is expected to be more prevalent in larger groups.

## Brief Instructions

We have set the default parameters to create an economic environment where public good provision is efficient, but voluntary contributions will likely be insufficient to reach the threshold.

1. Cost of public good as a fraction of group's endowment (Threshold (%)) is 50%. This means the good is provided if the average contribution is one-half of each player's endowment, and the "fair" equilibrium of each contributing half of her endowment is easy to identify.
2. With Group Size equal to 5 and Rate of Return (%) equal to 30%, total group benefit is 5×30%=150% of the project's cost. Provision is therefore efficient.
3. If contributions are insufficient, contributions are not refunded to contributors (Refund=no).
4. If contributions are sufficient, funds in excess of cost of provision are not returned to students (Rebates=no).

When provision is efficient, there are Nash equilibria in which the good is provided. In these equilibria, group contributions are exactly equal to the threshold, and nobody contributes more than her individual benefit (15 with default parameters). Each player prefers those equilibria where the threshold is reached to those equilibria where it is not. Of course, each player prefers contributing less of the threshold than more of the threshold, which likely leads to coordination failures.

## Key Treatment Variations

1. Allowing Chat will generally increase the likelihood of sufficient funding.
2. In the event that contributions are insufficient, fully refunding all contributions (Refund=yes) will make achieving the threshold easier.
3. In the event that contributions are sufficient, returning excess contributions in proportion to each student's percentage of total contribution (Rebate=yes) will make achieving the threshold easier.
4. Changing Group Size affects the cost and benefit of the public good in different ways. For example, if you double Group Size without changing any other parameters, societal cost doubles. Doubling group size also doubles group benefit as a fraction of societal cost, meaning societal benefit quadruples! If you double Group Size and halve Rate of Return (%), then both societal cost and societal benefit double.

## Results

In the results, you will see evidence of free riding. You are also likely to see improved coordination when groups are fixed for multiple periods, as groups will move towards either the contribute-nothing equilibrium or the equilibrium in which the cost of the public good is shared equally by all. We start by looking at the Multi Period Summary in the Go To drop down.

In the top table (Figure 1), you are likely to see a decrease in average contributions per group in later periods, consistent with an increase in free riding. In addition, you are likely to see an increase in equilibrium play (improved coordination) in later periods: a reduction in over-contributing means the maximum group-level contribution will likely get closer to the threshold, and the minimum group-level contribution will get closer to zero.

This group-level information across periods is presented in the chart on the left (Figure 2). The figure on the left presents a box-and-whiskers plot of individual contributions per period.

In the Go To drop-down menu, you can focus on a particular period. The rightmost column of the table (Figure 3) shows which groups collected sufficient contributions to fund the public good. You can point out that earnings are higher in those groups meeting the threshold. In addition, as you move from presenting early periods to later periods, you should see a bifurcation in group outcomes, with those groups meeting the threshold having little in excess contributions, and those groups not meeting the threshold tending to zero contributions.

The figure (Figure 4), groups individuals into one of five categories. As you move to later periods, you should see an increase in two categories: those who contributed nothing (the first category) and those whose contribution was exactly an even share of the public good's cost (the third category).