Kevin J. S. Zollman (2021). The theory of games as a tool for the espitemologist. Philosophical Studies 178 (4): 1381–1401.
Can purely epistemic situations be situations of strategic interaction? Most certainly yes, argues Kevin Zollman in a thought-provoking new paper. There is much in this paper that is fascinating, including multiple philosophical claims, suppositions, and arguments relevant to many disciplines. Most of these will be set aside here. The focus here will rather be on the relation between the first substantive part of the paper (section two) and the second (section three). Section two builds up a framework for generating strategic considerations in ‘purely’ epistemic situations. It also describes a specific class of situations that amount to Prisoner’s Dilemma games. Section three builds up a slightly different framework and generates, within this framework, further stylised classic games, such as a Stag Hunt and a Coordination game. (And a Prisoner’s Delight game.) There is one class of games that is not generated in this framework: Chicken games. (Zollman is well aware of this and notes it in a footnote, 1392n17.) I want to know where this difficulty comes from: is this a purely pragmatic difficulty, such that if one were to keep trying, one could generate Chicken games? Or is there a more fundamental difference between the kind of conditions that generate Stag Hunt, Coordination, and Prisoner’s Dilemma (and Delight) games, on the one hand, and Chicken games, on the other? It seems to me that it is rather the latter.
Here’s the strategy. To get at the root of this (possible) inconsistency, we’ll approach this from a more general level, thinking about the kind of general conditions that allow for various types of games. If it turns out that the conditions allowing for some class of games are inconsistent with the conditions allowing for a different class of games, we must conclude that generating both classes within the same set of conditions is impossible. Within Zollman’s framework in section two, this translates to: if the properties that joint accuracy functions must satisfy to generate one class of (Stag Hunt and Coordination) games are inconsistent with the properties that joint accuracy functions must satisfy to generate another class of (Chicken) games, then these two classes of games cannot be generated by the same class of joint accuracy functions. To understand this, we need some preliminaries explaining Zollman’s framework in section two. And then we need to generate, within that framework, the types of games generated in the paper within the framework of section three.
Individual epistemic decision-making
Here’re the bare preliminaries:
“Suppose a single proposition \(p\) which an agent assigns a credence \(c\). If \(p\) turns out to be true, our agent is assigned an accuracy score of \(S(c, 1)\) and if it turns out to be false \(S(c, 0)\). [\(\dots\)]
Our agent has credence \(c\) and she is evaluating the expected accuracy of alternative credence in the same proposition, \(c^{\prime}\). She expects \(c^{\prime}\) would receive the score: \(E(c^{\prime}, c) = cS(c^{\prime}, 1) + (1-c)S(c^{\prime}, 0)\).” (1384)
A key concept in this framework is the idea of a ‘proper scoring rule’. It is defined as follows:
Proper scoring rule. Let \(S(c, y)\) with \(c \in [0, 1]\) and \(y \in \{0, 1\}\) be a scoring rule. And let \(E(c^{\prime}, c)\) with \(c, c^{\prime} \in [0,1]\) be an expectation function that yields the expected accuracy of credence \(c^{\prime}\), that is, \(E(c^{\prime}, c) = E(c, S(c^{\prime}, 1), S(c^{\prime}, 0))\). Then \(S\) is a proper scoring rule if for all credences \(c\):
\(c = \text{argmax}_{c^{\prime} \in [0,1]} E(c^{\prime}, c)\)
This means that for all \(c\) and all \(c^{\prime} \neq c\), we have \(E(c, c) \geq E(c^{\prime}, c)\), with strict inequality yielding strict propriety. Or, intuitively, a scoring rule is proper if an agent’s credence maximises the agent’s expected accuracy, relative to that credence.
\(n\)-person epistemic decision-making
Now, all of the above describes evaluations by a single agent. So let’s introduce strategic considerations by introducing more agents. Let \(N = \{1, \dots, n\}\) be the set of agents. (The notation above will now be indexed by the respective agent \(i\).) Further, let vector \(\mathbf{x} = (x_{1}, \dots, x_{n})\) collect credences for each agent. Now, define the joint accuracy function of agent \(i\) to be a function of the expected accuracy that \(i\) assigns to each credence in \(\mathbf{x}\); that is, \(JA_{i}(\mathbf{x}, c_{i}) = JA_{i}(E_{i}(x_{1}, c_{i}), \dots, E_{i}(x_{n}, c_{i}))\).
With this machinery at hand, it is easy to derive a generalisation of Zollman’s Proposition 2. According to this proposition, ‘sticking to one’s own prior credence’ weakly dominates the strategy of adopting a compromise credence, given a linear joint accuracy function. In fact, it is easy to show that ‘sticking to one’s prior credence’ is a weakly dominant strategy for a large class of joint accuracy functions. (And hence weakly dominates not just adopting a compromise credence, but also adopting any other credence.)
As long as each agent is using a joint accuracy function that is monotonic in the agents’ expectation functions, then we have:
Suppose that, for all \(i \in N\), \(S_{i}\) is a strictly proper scoring rule. Then as long as \(JA_{i}\) is monotonic in \(E_{i}\) for all \(i \in N\), we have:
\(JA_{i}((c_{i}, \mathbf{x}_{-i}), c_{i}) \geq JA_{i}(\mathbf{x}, c_{i})\)
Where \((c_{i}, \mathbf{x}_{-i})\) is the vector of credences where \(i\)’s credence is \(c_{i}\) and everyone else’s is as in \(\mathbf{x}\).
(The inequality follows immediately from monotonicity, the fact that \(E(c_{i}, c_{i})\) and \(E(x_{i}, c_{i})\) are the only differing arguments, and the fact that, due to strict propriety, \(E(c_{i}, c_{i}) \geq E(x_{i}, c_{i})\), with a strict sign when \(x_{i} \neq c_{i}\).)
This more general result establishes that for a large class of monotonic joint accuracy functions, each agent has, for any possible credence the agent may hold, a weakly dominant strategy: namely, sticking to this prior credence. And this, in turn, implies that for any vector collecting the prior distribution of credences, this vector is a Nash equilibrium. That is to say, if \(\mathbf{c}\) is the vector of credences of the agents in \(N\), then \(\mathbf{c}\) is a Nash equilibrium.
The question now is: is \(\mathbf{c}\) a unique Nash equilibrium? The answer hinges on the kind of mononicity the joint accuracy functions satisfy. Let’s start from the strictest possible case. First, assume as before that \(S_{i}\) are strictly proper for all \(i\), so that there is a unique credence \(c\) that maximises \(E_{i}(c, c_{i})\): namely, \(c = c_{i}\). Then, consider the following property:
Strict monotonicity in own credence. Let \(JA_{i}(\mathbf{x}, c_{i})\) be \(i\)’s joint accuracy function. \(JA_{i}\) is strictly monotonic in \(i\)’s own credence if:
\(JA_{i}((y_{i}, \mathbf{x}_{-i}), c_{i}) > JA_{i}(\mathbf{x}, c_{i})\)
Whenever:
\(E_{i}(y_{i}, c_{i}) > E_{i}(x_{i}, c_{i})\)
If all joint accuracy functions are strictly monotonic in the agents’ own credence, then this is sufficient to yield a uniqueness result:
Unique Nash equilibrium. Take a set of agents \(N\) and a credence distribution \(\mathbf{c}\). For all \(i \in N\), let \(S_{i}\) be a strictly proper scoring rule and let \(JA_{i}\) be a joint accuracy function that is strictly monotonic in \(i\)’s own credence. Then \(\mathbf{c}\) is the unique Nash equilibrium.
The proof is immediate and follows from the fact that each \(c_{i}\) uniquely maximises \(E_{i}(\cdot, c_{i})\) and that \(JA_{i}\) is strictly monotonic in \(i\)’s own credence.
(Notice that strict monotonicity in own credence is quite weak: it only requires strict monotonicity with respect to the agent’s own credence rather than with respect to any possible credence, potentially those of other agents.)
Of course, situations with a unique Nash equilibrium need not be Prisoners’ Dilemma games. One would need further restrictions to ensure the existence of a different credence vector that Pareto dominates the equilibrium. (Put differently, Proposition 1 does not generalise as easily as Proposition 2. Perhaps this is why Zollman admits that the specific functional form of the joint accuracy function he works with is “not an innocuous assumption”, 1388.) What I am more interested in here, however, is the possibility of generating Chicken games. A Chicken game is a multiple-equilibria game where the equilibria involve agents taking different actions. Now, in the framework above, this is still a bit vague so we need to be a bit more careful about what the relevant actions are.
Generating multiple equilibria
Let’s start by distinguishing between two types of ‘actions’: one possible action is to stick to your given credence; another possible action (type) is to change your credence (with token actions here referring to all the credences one could pick). In a game with joint accuracy functions that are strictly monotonic in one’s credence, the unique Nash equilibrium is the vector of each agent sticking to their credence. (It is at places like these that the ‘epistemic versus pragmatic’ distinction Zollman opens the paper with starts to thread on shaky ground. I’ll bracket this issue as I don’t think it’s that crucial to the insights in the paper.) So if we want to generate multiple equilibria, we need to relax the assumption of strict monotonicity. Take weak monotonicity:
Weak monotonicity in own credence. Let \(JA_{i}(\mathbf{x}, c_{i})\) be \(i\)’s joint accuracy function. \(JA_{i}\) is weakly monotonic in \(i\)’s own credence if:
\(JA_{i}((y_{i}, \mathbf{x}_{-i}), c_{i}) \geq JA_{i}(\mathbf{x}, c_{i})\)
Whenever:
\(E_{i}(y_{i}, c_{i}) > E_{i}(x_{i}, c_{i})\)
Under the assumption of weak monotonicity, uniqueness no longer necessarily holds. The trouble is that while weak monotonicity allows for games with multiple equilibria, some of which are weak equilibria, it does not allow for games with multiple strict equilibria. And it is these games that Zollman generates later in the paper. Recall that a Coordination game and a Stag Hunt game (like a Chicken game) is a game with multiple strict equilibria: under different credence vectors, agents have different strict best responses. Weak monotonicity implies that under any credence vector, an agent always has the same (possibly weak or strict) best response: namely, sticking to their prior credence.
Let’s take the two-agent case. Suppose that the vector of prior credences is \(\mathbf{c}\) and that agents could change to an alternative vector \(\mathbf{x}\). Then, for \(i = 1, 2\), to allow for a Stag Hunt or a Coordination game, we need to allow for (of course, there’re further restrictions that delineate these games and make them unique; in any case, they’re both games with two strict equilibria):
\((1) \qquad JA_{i}(\mathbf{c}, c_{i}) > JA_{i}((x_{i}, \mathbf{c}_{-i}), c_{i})\)
\((2) \qquad JA_{i}(\mathbf{x}, c_{i}) > JA_{i}((c_{i}, \mathbf{x}_{-i}), c_{i})\)
In other words, to generate a Stag Hunt or a Coordination game, sticking to one’s own credence should be a strict best response to the other player sticking to their credence, and changing one’s own credence should be a strict best response to the other player changing their credence.
The second inequality clearly violates weak (and strict) monotonicity. But if we weaken that condition even further, then we can allow for such strict equilibria. Consider:
Strict monotonicity in own credence with respect to \(\mathbf{c}\). Let \(\mathbf{c}\) be the vector of given credences. Let \(JA_{i}(\mathbf{x}, c_{i})\) be \(i\)’s joint accuracy function. \(JA_{i}\) is strictly monotonic in \(i\)’s own credence with respect to \(\mathbf{c}\) if:
\(JA_{i}((y_{i}, \mathbf{x}_{-i}), c_{i}) > JA_{i}(\mathbf{x}, c_{i})\)
Whenever:
\(E_{i}(y_{i}, c_{i}) > E_{i}(x_{i}, c_{i})\) and \(\mathbf{x} = \mathbf{c}\)
This condition restricts strict monotonicity in one’s own credence to a unique vector: namely, the vector of prior credences. And it implies that this vector of prior credences is a strict Nash equilibrium. However, it does not rule out games where changing one’s credence is a strict best response to a different vector of credences. (Although notice that while this weaker condition allows for multiple strict equilibria, it is not sufficient to guarantee they exist.) So requiring that the joint accuracy functions satisfy this very weak vector-specific form of monotonicity would allow us to generate the Stag Hunt and Coordination games described in the second part of the paper.
Generating a Chicken game
Now, Chicken games are games of multiple strict equilibria. Does the condition allow for them? Unfortunately, no. To see why, take the two-agent case described above and consider the conditions describing the two equilibria in a Chicken game:
\((3) \qquad JA_{i}((x_{i}, \mathbf{c}_{-i}), c_{i}) > JA_{i}(\mathbf{c}, c_{i})\)
\((4) \qquad JA_{i}((c_{i}, \mathbf{x}_{-i}), c_{i}) > JA_{i}(\mathbf{x}, c_{i})\)
In other words, to generate a two-player Chicken game, changing one’s own credence should be a strict best response to the other player sticking to their credence, and sticking to one’s own credence should be a strict best response to the other player changing their credence. Clearly, these two conditions are inconsistent with conditions \((1)\) and \((2)\) above. Put differently, joint accuracy functions that allow for Stag Hunt and Coordination games cannot allow for Chicken games.
Notice further that the weak monotonicity condition that allows for Stag Hunt and Coordination games, also allows for Prisoners’ Dilemma games. These three types of games can be generated using the same broad class of joint accuracy functions, given strictly proper underlying scoring rules. (With additional conditions delineating these games more narrowly.) But, given strictly proper underlying scoring rules, to generate a Chicken game, one needs to use non-monotonic joint accuracy functions. To see why, notice that condition \((3)\) requires one’s joint accuracy function to be decreasing in one’s expected credence when the other player sticks to their own credence: given \(\mathbf{c}\), a higher credence (\(E(c_{i}, c_{i})\)) should yield a lower joint accuracy than a lower credence (\(E(x_{i}, c_{i})\)) does. Further, condition \((4)\) requires one’s joint accuracy function to be increasing in one’s expected credence when the other player changes their own credence: given \(\mathbf{x}\), a higher credence (\(E(c_{i}, c_{i})\)) should yield a higher joint accuracy than a lower credence (\(E(x_{i}, c_{i})\)) does. This points to an interesting asymmetry between the joint accuracy functions that generate Stag Hunt and Coordination games, on the one hand, and Chicken games, on the other.
Strategic complements and substitutes
Here is a more intuitive way of thinking about this asymmetry. Consider again the running two-player example. A joint accuracy function that generates a Stag Hunt or a Coordination game must be such that it makes the agents’ actions strategic complements: if agent two changes their credence, that should make changing one’s credence (strictly) more appealing for agent one as well. In contrast, Chicken games are situations where the actions of the two agents are strategic substitutes: if agent two changes their credence, that should make changing one’s credence (strictly) less appealing for agent one. Clearly, joint accuracy functions that make the agents’ actions strategic complements cannot also make them strategic substitutes. (Except in the uninteresting degenerate case.) And thus functions that generate Stag Hunt and Coordination games cannot generate Chicken games.
Intuitively, this is a substantial difference. Joint accuracy functions that lead to strategically complementary actions describe agents who have some kind of prior epistemic trust; while those that lead to strategically substitute actions pertain to agents who are inherently epistemically distrustful of each other. Let’s look at examples of each of these functions.
Here is an example of a general joint accuracy function that can generate a Prisoner’s Dilemma, a Stag Hunt, or a Coordination game, depending on the specific functional form of the expected accuracy function:
\((5) \qquad JA^{C}_{1}((x_{1}, x_{2}), c_{1}) = E_{1}(x_{1}, c_{1}) \cdot H^{*}(E_{1}(x_{2}, c_{1}) - E_{1}(c_{2}, c_{1}))\)
Where \(H^{*}(z)\) is a version of the Heaviside function that yields a value of \(1\) whenever \(z \geq 0\) and a value of \(-1\) otherwise.
And here is an example of a joint accuracy function that can generate a Chicken game (again depending on the underlying expected accuracy function), but not the other three canonical games above:
\((5) \qquad JA^{S}_{1}((x_{1}, x_{2}), c_{1}) = E_{1}(x_{1}, c_{1}) \cdot G^{*}(E_{1}(x_{2}, c_{1}) - E_{1}(c_{2}, c_{1}))\)
Where \(G^{*}(z)\) is a version of the Heaviside function that yields a value of \(-1\) whenever \(z \geq 0\) and a value of \(1\) otherwise.
Notice that the only difference between \(JA_{i}^{C}\) and \(JA_{i}^{S}\) is the step function assigning a positive or negative value to the difference in the brackets. The difference in the brackets is the difference between the expected accuracy of the other agent changing their credence to the credence under consideration (\(x_{2}\)) relative to the status quo (\(c_{2}\)). In other words, the difference in the brackets is the ‘surplus’ expected accuracy from the other agent changing their credence to \(x_{2}\) (which, notice, could be \(c_{2}\), in which case the difference is zero). \(JA_{i}^{C}\) assigns a nonnegative ‘surplus’ a positive value, while \(JA_{i}^{S}\) assigns a nonnegative ‘surplus’ a negative value. Intuitively, this means that when the other agent improves their expected accuracy relative to the status quo, a \(JA_{i}^{C}\)-type of player would evaluate this positively, while a \(JA_{i}^{S}\)-type of player would evaluate this negatively. This differential treatment can be interpreted in various ways but it does point to a fundamental difference in the types of agents that can find themselves in a Stag Hunt or a Coordination game, on the one hand, or a Chicken game, on the other.
The idea of epistemic strategic complementarity and substitutability is also philosophically interesting (and, in its epistemic form, as far as I know, novel), particularly with respect to its relation to issues of disagreement, and epistemic cooperation. Complementarity leads to mutual epistemic reinforcement while substitutability leads to mutual epistemic divergence, and it is the latter case that might be most costly, not just in terms of the difficulty of overcoming disagreement but also in terms of its conduciveness to polarization.