The Cooperative Origins of Epistemic Rationality?

Abstract

Recently, both evolutionary anthropologists and some philosophers have argued that cooperative social settings unique to humans play an important role in the development of both our cognitive capacities and what Michael Tomasello terms the “construction” of “normative rationality” or “a normative point of view as a self-regulating mechanism.” In this article, I use evolutionary game theory to evaluate the plausibility of the claim that cooperation fosters epistemic rationality. Employing an extension of signal-receiver games that I term “telephone games,” I show that cooperative contexts work as advertised: under plausible conditions, these scenarios favor epistemically rational agents over irrational ones designed to do just as well as them in non-interactive contexts. I then show that the basic results are strengthened by introducing complications that make the game more realistic.

Appendix: Technical Details

1.1 Defining Agential Behavior

Let $Pr_s$ indicate the agent’s subjective probability, $Pr_o$ the objective probability, and u the utility function for that agent. Our rational agents have perfect uptake of the objective probabilities (i.e., for all $s_i \in S$, $Pr_s(s_i|o) = Pr_o(s_i|o)$), act to maximize expected utility, and are “credulous” in the sense that when they receive a report regarding the probability distribution from another agent, their subjective probability assignment is the same as the assignment that they receive (letting $Pr_r$ indicate the probability assignment received: for all $s_i \in S$, $Pr_s(s_i) = Pr_r(s_i)$). Credulity is an idealization, obviously, but it may be a good one (Zollman, 2015).

Wishful agents are little more complicated. First, they assign probabilities as follows:

$$\begin{aligned} Pr_s(s_i|o)&= \frac{Pr_o(s_i|o)u(s_i)}{\sum _{s_j \in S} Pr_o(s_j|o)u(s_j)} \end{aligned}$$

which is to say that their probability assignment is equivalent to their expected utility for each state normalized so as to behave like probabilities. Similarly, when receiving a probability assignment, they incorporate their utilities into their assignment of probabilities like so:

$$\begin{aligned} Pr_s(s_i)&= \frac{Pr_r(s_i)u(s_i)}{\sum _{s_j \in S} Pr_r(s_j)u(s_j)} \end{aligned}$$

In other words, wishful agents are “credulous” in the same sense that rational agents are: they don’t discount the reports of others by taking into account the trustworthiness of the sender or by or incorporating their own priors. Instead, they take the probabilities at “face value,” only incorporating their preferences, just as they do in response to an observation. Finally, in order to avoid double-counting their utilities, they determine actions a little bit differently, namely by picking the action that has the highest “probability” according to their assignment.

Note that this particular way of cashing out what “wishful” means only results in identical behavior with rational agents in non-interactive scenarios due to stipulations about the payoff structure. In particular, it requires that action x and state x are guaranteed to have the same expected utility.

1.2 Determining Expected Utilities

In any given scenario, every agent has an expected utility. So, for example, in a degree-2 cardinality-2 game with one rational agent and one wishful agent, mean observation power set to .75, and observation power $\sigma ^2$ of .05, the expected utility of the rational agent ($u_R$), wishful agent ($u_W$), and actor ($u_A$) are $u_R(1,1) = 1.19$, $u_W(1,1) = 1.29$, and $u_A(1,1) = 1.29$, where the numbers in parentheses indicate the number of rational and wishful agents in the chain respectively. To determine these values, I ran one million simulations for each scenario.

The overall expected utility for an agent is then determined by how likely they are to find themselves in each possible scenario, which depends on the agent themselves and the proportion of agents playing the rational strategy. Letting R indicate this latter value, the overall expected utility for a rational agent in any degree-2 cardinality-2 game is

$$\begin{aligned} u_R&= \frac{2}{3}\Big [\big [R * u_R(2,0)\big ] + \big [(1-R) * u_R(1,1)\big ]\Big ] \nonumber \\&\quad +\,\frac{1}{3}\Big [\big [R^2 * u_A(2,0)\big ] + \big [2* R*(1-R)* u_A(1,1)\big ] \nonumber \\&\quad +\,\big [(1-R)^2 * u_A(0,2)\big ]\Big ] \end{aligned}$$

(1)

Spelling this out: any agent in a cardinality-2 game has a 2 in 3 probability of being a testifier rather than an actor; given that they are a testifier, their probability of being paired with a rational agent is given by R and their probability of being paired with a wishful agent by $1-R$. These facts allow us to weight the probability of different scenarios that the rational agent can find themselves in, giving us their total expected utility when entering into the game. The expected utility for a wishful agent, while identical in the second term, differs from the above in that the first term is

$$\begin{aligned} u_W&= \frac{2}{3}\Big [\big [R * u_W(1,1)\big ] + \big [(1-R) * u_W(0,2)\big ]\Big ] + \cdots \end{aligned}$$

(2)

The difference stems from the features of the individual agent themselves: wishful agents can’t find themselves in a group of only rational agents, because they themselves are part of the group (and similarly for rational agents).

1.3 The Initial Model

To run the model itself, I set R to .5 and then allowed it to evolve according to a discrete version of the replicator dynamics. That is, at each step n, $R(n+1)$ was calculated by adding the weighted advantage of rational agents over the average agent like so:

$$\begin{aligned} R(n+1)&= R(n) + R(n)\big [u_R - [R(n)*u_R + (1-R(n))*u_W]\big ] \end{aligned}$$

(3)

The simulation halted when one of three conditions was met: (a) one strategy was played by $99.99\%$ or more of agents, (b) the percentage of agents playing each strategy after a given round was identical (to the hundredth of a percent) to the percentage playing after the previous round, (c) 1000 rounds had passed. (Why 1000 rounds? I found that there are three different behaviors that I wanted to distinguish: (a) halting at a percentage substantially under 100%, (b) relatively quick but ultimately asymptotic approach of 100%, and (c) very slow movement towards 100%. At 500, it’s hard to distinguish between (a) and (c). At 1500 it starts to get hard to distinguish between (b) and (c). 1000 is a happy medium.)

A summary of the results for the first game can be found in Table 2.

Table 2 Summary of simulation results for games of cardinality 2, 3, and 4, where “$\%$R” indicates the percentage of the population playing the rational strategy and “Turns” the number of turns the model took to shut off

1.4 Complicating the Model

The complications introduced in Sect. 5 work as follows. First, I introduced non-cooperators who do not interact with other agents: they simply take a guaranteed payoff, which ensures that their expected utility is equal to 1 + 1/the cardinality of the game (the second term represents their chance of getting a “preferred” state via this guaranteed payoff). Second, I had cooperators of both wishful and rational variety randomly attempt to cooperate with five other individuals, with the degree of the game depending on how many of those individuals were also cooperators and failure to find any other cooperators resulting in a payoff of 0. This captures the element of risk associated with the cooperative strategy: the more non-cooperators in the population, the higher the probability a cooperator gets nothing. Third, to capture the benefits of cooperation, I increased the payoffs depending on the number of cooperators in the group: it’s hard to get five cooperators together—and doing so risks massive information loss—but the rewards for success in such cases are dramatic. Specifically, I set the advantage equal to 1 + the number of total cooperators in the interaction divided by 2. (This particular choice of advantage is largely arbitrary, but the overall pattern of when cooperation is favored doesn’t depend on it.) The expected utility of a rational agent is then given by:

$$\begin{aligned} u_R&= \big [Pr(1\ \text {cooperator}) *\ \text {ex. payoff in d-1 game} *\ \text {adv. of d-1 game}\big ]\nonumber \\&\quad + \big [Pr(2\ \text {cooperators}) *\ \text {ex. payoff in d-2 game} *\ \text {adv. of d-2 game}]\ +\cdots \end{aligned}$$

(4)

$$\begin{aligned}= \big [5(1-N)(N)^4 * u_{R_1} * 1.5\big ]\ +\ \big [10(1-N)^2(N)^3 * u_{R_2} * 2\big ]\ +\cdots \end{aligned}$$

(5)

where N is the proportion of the population who are not cooperating and $u_{R_x}$ (or, in the wishful case, $u_{W_x}$) the expected utility of agents in games of degree-x. The extension to the wishful case is straightforward.

Introducing the additional kind of agent required slightly altering the dynamics. Rather than (3), above, $R(n+1)$ and $N(n+1)$ were calculated like so:

$$\begin{aligned} R(n+1)= R(n) + (1-N(n))*R(n)\big [u_R - [R(n)*u_R + (1-R(n))*u_W]\big ] \end{aligned}$$

(6)

$$\begin{aligned} N(n+1)&= N(n) + N(n)\Big [u_N - \big [N(n)u_N + (1-N(n)) \nonumber \\&\quad *[R(n)*u_R + (1-R(n))*u_W]\big ]\Big ] \end{aligned}$$

(7)

The second equation is standard: $N(n+1)$ is calculated by adding the weighted advantage of non-cooperators over the average agent. The first is modified to account for the assumption that the difference between rational and wishful agents is only relevant to cooperators.

A summary of the results for the second game can be found in Table 3.

Table 3 Summary of simulation results for the more complicated games. As before, %R indicates the percentage of rationals, %N indicates the percentage of non-cooperators. The initial proportion of rational to wishful agents in all three cardinalities was 1:1