The Free-Rider Problem: A Social Dilemma

What is a Public Good?

First, let's clarify what we mean by a "public good." It's a resource or service with two key properties:

1. Non-excludable: It is difficult or impossible to prevent anyone from using the good, even if they didn't contribute to its creation.
2. Non-rivalrous: One person's use of the good does not diminish its availability to others.

Let's watch a short video that provides some concrete examples and explains why these properties naturally lead to a strategic problem.

As the video explains, examples include city parks, clean air, national defense, and open-source software. The core tension is that since everyone can benefit regardless of their contribution, there is a powerful incentive to let others bear the cost. This is known as the free-rider problem.

This situation is closely related to the "Tragedy of the Commons," a concept you might have encountered. The key distinction is that common-pool resources (like grazing land or fisheries in the classic examples) are typically rivalrous (one person's use depletes the resource for others) but still non-excludable. The public goods we'll model today are non-rivalrous.

Modeling the Public Goods Game

To analyze this problem, we use a model called the Public Goods Game, or in its experimental form, the Voluntary Contribution Mechanism (VCM).

Let's start with a visual walkthrough of how the game works.

Now, let's use our PAIP framework to build a formal model of this game. Since you appreciate mathematical rigor, we'll define the payoffs with a precise formula.

• Players: A group of $N$ individuals, indexed by $i=1,...,N$.
• Actions: Each player $i$ is given an initial endowment, $e_i$. They must decide how much of this endowment, $x_i$, to contribute to a public pot, where $0\le x_i\le e_i$. The amount they don't contribute, $e_i-x_i$, they keep for themselves.
• Information: All players choose their contribution $x_i$ simultaneously, without knowing the choices of others. This is a game of imperfect information.
• Payoffs: The total contributions in the pot are multiplied by a factor $k>1$. This new, larger pot is then divided equally among all $N$ players, regardless of their individual contributions.

The payoff for player $i$, denoted ${\pi }_i$, is the sum of what they kept and their share of the public pot:

${\pi }_i(x_1,...,x_N)=(e_i-x_i)+\frac{k}{N}{\sum }_{j=1}^N{x}_j$

Let's define the term $m=k/N$ as the Marginal Per Capita Return (MPCR). This is the personal return player $i$ gets for each single unit contributed to the pot (by anyone, including themselves). Our payoff function becomes:

${\pi }_i(x_1,...,x_N)=(e_i-x_i)+m{\sum }_{j=1}^N{x}_j$

The social dilemma arises under two specific conditions on the MPCR:

1. $m<1$: The individual return from contributing one unit is less than the cost of contributing that unit.
2. $m\times N>1$ (which is equivalent to $k>1$): The total return to the group from one contributed unit ($m\times N=k$) is greater than the cost of that unit.

Identifying the Free-Rider Problem Mathematically

With this model, we can now precisely identify the incentive to free-ride. Let's analyze the marginal effect of player $i$'s contribution on their own payoff.

Consider what happens if player $i$ decides to increase their contribution by one small unit, $\Delta x_i$.

• The amount they keep, $(e_i-x_i)$, decreases by $\Delta x_i$.
• The public pot, $\sum x_j$, increases by $\Delta x_i$.
• Their share of the pot, $m\sum x_j$, increases by $m\times \Delta x_i$.

The net change in player $i$'s payoff is $-\Delta x_i+m\Delta x_i=(m-1)\Delta x_i$.

Since the condition for the social dilemma is $m<1$, the term $(m-1)$ is negative. This means that for any increase in their contribution, player $i$'s payoff decreases. Therefore, a rational player seeking to maximize their own payoff will choose the lowest possible contribution: $x_i=0$.

This is the free-rider problem in its starkest form. Contributing nothing is a strictly dominant strategy for every player. The Nash Equilibrium is for all players to contribute zero.

The following video explains this logic using a specific example.

The N-Player Social Dilemma

We have established that the individually rational strategy is to contribute nothing. But is this outcome good for the group?

• Nash Equilibrium Outcome: If everyone plays their dominant strategy, $x_i=0$ for all $i$. The total contribution is 0. Each player's payoff is ${\pi }_i=(e_i-0)+m(0)=e_i$.
• Socially Optimal Outcome: If everyone contributes their full endowment, $x_i=e_i$ for all $i$. Each player's payoff is ${\pi }_i=(e_i-e_i)+m{\sum }_{j=1}^N{e}_j=m\cdot N\cdot e$. (Assuming all endowments $e_i$ are equal to $e$).

Recall our second condition for the dilemma: $m\times N>1$. This means $mNe>e$. Every single player is better off in the world of full cooperation than in the world of zero cooperation.

This conflict is the essence of a social dilemma: the outcome produced by individually rational choices (the Nash Equilibrium) is worse for everyone than an alternative, cooperative outcome. The Nash Equilibrium is Pareto inefficient.

This structure is a generalization of the 2-player Prisoner's Dilemma to $N$ players, and is often called the N-Person Prisoner's Dilemma (NPD). The academic paper below provides the formal mathematical treatment of this connection.

The key insight from this model is that the "tragedy" of non-cooperation isn't caused by ignorance or malice. It's a direct consequence of a strategic structure where individual incentives are misaligned with collective goals.

Beyond the Basic Model

It's worth noting that the linear model we've analyzed, where each contributed unit has the same impact, is the most pessimistic version of the public goods game. Many real-world public goods have non-linear payoffs. For instance, some projects require a minimum threshold of contribution before they provide any benefit at all (e.g., building a bridge).

In such non-linear games, the strategic analysis becomes more complex. It's no longer always a dominant strategy to contribute nothing. As explained in the paper by Archetti and Scheuring (in sections you don't need to read for this lesson), these more realistic models can have equilibria where a mix of cooperators and defectors coexist, even in a one-shot game. This shows that while the free-rider problem is a powerful force, the exact structure of the payoffs determines whether it leads to a complete breakdown of cooperation.

Conclusion

In this lesson, we applied our modeling skills to a multi-player environment. We have seen how to formally model a public goods game and how its strategic structure inevitably leads to the free-rider problem.

Our key takeaways are:

• A public good is defined by being non-excludable and non-rivalrous.
• We can model the contribution problem using the Voluntary Contribution Mechanism (VCM), where each player's payoff is ${\pi }_i=(e_i-x_i)+m\sum x_j$.
• When the individual return ($m$) is less than 1 and the group return ($m\times N$) is greater than 1, a social dilemma occurs.
• The dominant strategy for every player is to contribute nothing ($x_i=0$), leading to a Pareto inefficient Nash Equilibrium where everyone is worse off than if they had all cooperated. This is the N-Person Prisoner's Dilemma.

This model provides a powerful lens for understanding a vast range of real-world problems, from funding public services and environmental protection to teamwork in an office and the viability of open-source projects.

Preview of the Next Lesson:

We will continue our exploration of counter-intuitive outcomes in strategic interactions. Next, we will analyze Braess's paradox in traffic routing, a fascinating and surprising example of coordination failure where adding a resource (a new road) can make everyone's outcome worse.