## Public goods and prisoner’s dilemma games are usually equivalent

This is a old note showing that Choi and Bowles’ [CB07] public goods (PG) game is equivalent to the classic prisoner’s dilemma (PD) game.

### Public goods

To summarize, Choi and Bowles’ PG (and most other PGs) is as follows:

1. each cooperator contributes some amount $c$ to the public pool
2. all contributions are summed together and multiplied by some constant factor $r$ to get a public good of value $b$.
3. the public good $b$ is divided among all $n$ members (both cooperators and defectors) of the group equally, resulting in each agent receiving $\frac{b}{n}$.

From this definition, and the assumption that $k$ of the $n$ group members are cooperators ( $n - k$ are defectors), we can arrive at the following payoffs for cooperators ( $P_C$) and defectors ( $P_D$): $P_C = \frac{b}{n} - {c}$ $P_D = \frac{b}{n}$

We know that $b$ is the sum of all contributions ( $kc$) times some constant $r$. Therefore, $b = rkc$, and we can rewrite the previous equations as: $P_C = \frac{rkc}{n} - {c} = \frac{c}{n}(rk - n)$ $P_D = \frac{rkc}{n} = \frac{c}{n}(rk)$

### Prisoner’s dilemma

A popular formulation of prisoner’s dilemma is based on the cost of giving ( $\gamma$) and benefit of receiving ( $\beta$) (Pardon the Greek notation, I want to reserve $c$ and $b$ for PG). In this version, when an agent cooperates, she pays a cost $\gamma$ in order to give a benefit $\beta$ to the person she is interacting with; a defector pays nothing and gives nothing to his partner. If we have $n$ agents interacting (such that each agent plays a PD with each other agent, and themselves), and of those $n$ agents, $k$ are cooperators then we can construct the payoffs for cooperators ( $P_C$) and defectors ( $P_D$) as follows: $P_C = k\beta - n\gamma$

A cooperator receives a benefit from every cooperator ( $k\beta$) but also pays a cost to every agent she interacts with ( $n\gamma$). A defector on the other hand, only benefits and pays no cost: $P_D = k\beta$

Now, if we set the cost of cooperation $\gamma = \frac{c}{n}$, and the benefit $\beta = \frac{rc}{n}$ and substitute into the above equations, we get: $P_C = k\frac{rc}{n} - n\frac{c}{n} = \frac{c}{n}(rk - n)$ $P_D = k\frac{rc}{n} = \frac{c}{n}(rk)$

### Equivalence

Now, if we simply look at the last two equations for PG and PD, we might conclude equality. However, there is a subtle distinction. The PG game can be characterized by two constant parameters $c$ and $r$; PD is characterized by one constant parameters ( $r$) and a dynamic one that depends on group population, $\frac{c}{n}$. A dynamic game can potentially add an extra level of unneeded complexity, however, I will show that in the case of Choi and Bowles PG (and most PG-type games) there is no extra complexity added by interpreting the game as a PD.

The reason that no complexity is added, is in how payoffs are interpreted In Choi and Bowles’ own words:

they reproduce in proportion to their share of the group’s total payoffs

When something is `in proportion’, that means if there is a common factor that is shared by all individuals, then it can be set to $1$. As we can see from our presentation of the PG and PD equations, all individuals share $\frac{c}{n}$ in common; thus we can replace it by $1$ without effecting the dynamics of reproduction (which is what we really care about). Thus, the PG/PD equations become: $P_C = rk - n$ $P_D = rk$

And in the PD game we are safe to replace the cost of cooperating by $1$ (instead of $\frac{c}{n}$), thus eliminating the variable parameter without effecting the dynamics of reproduction. Thus PD and PG really are equivalent, and we are safe to apply our knowledge of how prisoner’s dilemma games are played to the Choi and Bowles simulations.

The only subtle point left, is the presence of self-interaction. As we will see in a future post, that can prove to be a surprisingly powerful feature.

### References

[CB07] J.-K. Choi and S. Bowles  “The Coevolution of Parochial Altruism and War” Science 318(5850): 636-640