Replicator dynamics of perception and deception

In this note we study the replicator dynamics of perception and deception. In particular, we consider a variant of the prisoners dilemma where we have 4 basic strategies, cooperators C, defectors D, perceivers CI, and deceivers DI.

Model 1

Cooperators pay a cost c to provide a benefit b to any other agent. Defectors never pay a cost, but receive benefit from those that donate. Perceivers can detect defectors and thus cooperate only with cooperators, perceivers, and the tricky deceivers. For this perceptive ability, they past an extra cost k. Deceivers are like defectors, except they are capable of tricking even perceivers into cooperating with them. This deception carries a cost l. Letting p be the proportion of cooperators, q for perceivers, and r for defectors (thus the proportion of deceivers is 1 - p - q - r), we can write down the utility
functions for the 4 strategies:

\begin{aligned}  U(C) & = b(p + q) - c \\  U(CI) & = b(p + q) - c(1 - r) - k \\  U(D) & = bp \\  U(DI) & = b(p + q) - l  \end{aligned}

Note that as long as k,l > 0, the only pure evolutionary stable strategy is to defect. However, in the case of 0 \leq c \leq l \leq k + c, the invasion graph is non-trivial. In particular, a world of

  • CI can be invaded by DI and C, but not by D.
  • DI can be invaded by C or D.
  • C can be invaded by D.
  • D is ESS

This suggests that there is potential for an internal fixed point. Let us equate the utilities to find the fixed point:

  • Setting U(C) = U(CI) we get r = k/c
  • Setting U(C) = U(D) we get q = c/b
  • Setting U(C) = U(DI) we get c = l

Unfortunately, c = l means that this fixed-point is a knife-edge phenomena in the larger parameter space of (b,c,k,l). Thus, let us consider the parameters 0 \leq k \leq c = l \leq b (the last inequality comes from the definition of PD). Thus, the dynamics keep a whole line fixed, parametrizing by p, we get the following stable distribution of strategies:

(p \;, \; k/c \;, \; c/b\; ,\; 1 - k/c - c/b - p)

Note that this places further constraints on our parameters, since we need k/c + c/b \leq 1 or $kb + c^2 \leq bc$. Thus, the parameter range is reduced to 0 \leq k \leq c(1 - \frac{c}{b}) and 0 < c = l. Note that in the stable distribution the highest social welfare is when there is no DI agents, and it is reasonable to believe that the population would converge to this point on the C-CI-D face of the strategy simplex. Let us compute the stability criterion:

\begin{aligned} s \cdot P s^* & = \frac{1}{bc}\begin{pmatrix} x & y & 1 - x - y \end{pmatrix} \begin{pmatrix} b - c & b - c & - c \\ b - c - k & b - c - k & - k \\ b & 0 & 0 \end{pmatrix} \begin{pmatrix} bc - bk - c^2 \\ bk \\ c^2 \end{pmatrix} \\ & = \frac{1}{bc}\begin{pmatrix} x & y & 1 - x - y \end{pmatrix} \begin{pmatrix} (b - c)(bc - c^2) - c^3 \\ (b - c - k)(bc - c^2) - c^2k \\ b(bc - bk - c^2) \end{pmatrix} \\ & = \frac{b(cb - c^2) - x(bc^2 - b^2k) - y(c(bc - c^2) + bck - b^2k)}{bc} \\ & = b - c/b - x(c - bk/c) - y(c(1 - c/b) + k(1 - b/c)) \\ \end{aligned}

This shows the stable point is not ESS.

Model 2

From what we learned in model 1, we can define an alternative 3 strategy model which has even more interesting dynamics. Let cooperators pay a cost c_1 to give a benefit b to cooperators, perceivers, and defectors. Let perceivers pay a cost c_2 > c_1 to give a benefit b to cooperators, and perceivers, but not to defectors. Lastly, let defectors pay no cost to give no benefit. Thus, the extra cost perceivers must pay in order to distinguish agents that cooperate from those that don’t is captured in the higher cost of cooperation. For now, we will not worry about deceivers, and write down the utility functions with p as the proportion of cooperators, and q the proportion of perceivers.

\begin{aligned}  U(C) & = b(p + q) - c_1 \\  U(CI) & = (b - c_2)(p + q) \\  U(D) &= bp  \end{aligned}

Note that this game has no pure equilibria, and an interesting, rock-paper-scissors like invasion graph. In particular, a world of

  • CI can be invaded by C but not by D,
  • C can be invaded by D but not by CI, and
  • D can be invaded by CI but not by C.

From this invasion graph we can see that there must be an internal fixed point.

  • Setting U(C) = U(CI) we get p + q = c_1/c_2.
  • Setting U(C) = U(D) we get q = c_1/b

Thus, our fixed point is given by:

(c_1/c_2 - c_1/b\; ,\; c_1/b\; ,\; 1 - c_1/c_2)

The dynamics are then orbits around this fixed point. A nice choice of parameters is mutliplies of (b,c_1,c_2) = (6,2,3).

It is possible to add DI agents to this model, and we have two choices on how to penalize them. If we want DI to be invaded by D then we should penalize DI by a constant offset, like in Model 1. If we want to have D to DI drift then we should use ideas similar to how CI agents are different in Model 2. This means that we can have C, CI, and D agents receiving benefit b_1 while DI agents are penalized by receiving a benefit b_2 < b_1. I would lean towards the first case, since it is more general.