§1 · Introduction
Two questions a player can ask
Nash
“Given what the others are doing, what is my best response?”
Each player optimises alone. In the prisoner's dilemma this leads to mutual defection, even though everyone would be better off cooperating.
Kant · Roemer
“Which strategy would I want everyone to adopt?”
The player judges a strategy as if it were a universal law. Cooperation becomes rational without outside enforcement or altruism.
§2 · Kantian equilibria (Roemer, 2019)
Three ways to universalise
Simple Kantian equilibrium (SKE)
A strategy that every player prefers to see adopted by all. Every game with a common diagonal has one (Roemer, Prop. 2.1).
Multiplicative equilibrium $K_\times$
$$V^i(s_1,\dots,s_n)\ \ge\ V^i(rs_1,\dots,rs_n)\quad \forall i,\ \forall r\ge 0$$
Nobody wants to scale everyone's effort by the same factor. Example: cutting emissions proportionally.
Additive equilibrium $K_+$
$$V^i(s_1,\dots,s_n)\ \ge\ V^i(s_1+r,\dots,s_n+r)\quad \forall i,\ r\ge -s_i$$
Nobody wants to shift everyone's effort by the same amount. Example: “what if everyone paid a little more tax?”
§3 · Dynamic setting
Finite symmetric games
The paper restricts itself to symmetric two-player games with $n$ pure strategies. The game is fully described by a matrix $A$ (the other player receives $A^\top$). The population state is a point of the simplex $x\in\Sigma^n$, where $x_i$ is the share of players using $i$.
The starting point is the replicator:
$$\dot x_i = x_i\big[(e_i,Ax)-(x,Ax)\big]$$
Any candidate dynamic $\dot x_i = f_i(x,A)$ must satisfy two conditions:
$$\textstyle\sum_i f_i(x,A)=0 \ \ \text{(invariance)}\qquad f_i(x,A)=0\ \text{if}\ x_i=0\ \ \text{(boundary)}$$
The first keeps the population on the simplex. The second stops an absent strategy from appearing.
§3.1 · The Moralisator equation
Look at the diagonal, not the row
| Defect | Cooperate |
|---|
| Defect | −10 | 0 |
|---|
| Cooperate | −15 | −1 |
|---|
The replicator grows strategy $i$ according to $(Ax)_i$, the average payoff of $i$ against the current population. A Kantian player looks at $A_{ii}$ instead: what they would earn if everyone played $i$. Here defection pays $-10$ once it is universal, cooperation pays $-1$.
$$\dot x_i = x_i\Big[A_{ii}-\textstyle\sum_j x_jA_{jj}\Big]$$
Subtracting $\mathrm{Tr}(A)/n$ would not work: the equation would no longer depend on $x$ and would break conditions (1) and (2). The reference is therefore the population-weighted mean of the diagonal payoffs.
§3.2 · Properties
Three theorems
Theorem 1. The Moralisator satisfies invariance and the boundary condition; its solutions exist and are unique for all time (polynomial field, compact simplex).
Theorem 2. The mean diagonal payoff $\bar a(t)=\sum_j x_j(t)A_{jj}$ is non-decreasing.
$$\frac{d\bar a}{dt}=\sum_j x_j\big(A_{jj}-\bar a\big)^2\ \ge 0$$
This is the variance of the diagonal payoffs under $x$, an analogue of Fisher's fundamental theorem.
Theorem 3. If $A_{11}>A_{ii}$ for all $i\ne1$ and $x_1(0)>0$, then $x_1(t)\to1$. A moralising population converges to the best universal strategy.
If several strategies share the diagonal maximum, the solution converges to a mixture of them, which is structurally unstable.
§3.3 · Interpolation
The morality coefficient $k$
No population is purely selfish or purely moral. The k-Replimorator equation is the convex combination of the two fields:
$$\dot x_i = x_i\Big(k\Big[A_{ii}-\textstyle\sum_j x_jA_{jj}\Big]+(1-k)\Big[\textstyle\sum_j x_jA_{ij}-\sum_{j,l}x_jx_lA_{jl}\Big]\Big)$$
It inherits invariance and the boundary condition. For a two-strategy game, with $x$ the share of the tracked strategy, it reduces to
$$\dot x = x(1-x)\,g_k(x),\qquad g_k(x)=k\,\Delta_{\text{diag}}+(1-k)\,\Delta_{\text{payoff}}(x)$$
$g_k$ is affine in both $x$ and $k$, which is what yields the closed-form thresholds $k_c$ of §4. The lab computes these thresholds directly from whatever matrix you enter.
§4.4 and §5 · What the examples show
Three regimes, depending on the equilibria
Pure ESS: a threshold switch
Prisoner's dilemma, stag hunt. Below $k_c$ the population heads to the bad equilibrium, above it to the Pareto optimum. The threshold drops as initial cooperation rises.
Mixed ESS: a continuous shift
Hawk–dove. Morality pushes the mixed equilibrium towards more doves, up to pure dove.
Coordination: a growing basin
Cow, stag and hare. The optimum's basin grows with $k$ and covers almost the whole simplex around $k=0.3$.