Abstract
Whereas The Stag Hunt and the Evolution of Social Structure supplements Evolution of the Social Contract by examining some of the earlier work’s strategic problems in a local interaction setting, no equivalent supplement exists for The Dynamics of Rational Deliberation. In this article, I develop a general framework for modeling the dynamics of rational deliberation in a local interaction setting. In doing so, I show that when local interactions are permitted, three interesting phenomena occur: (a) the attracting deliberative equilibria may fail to agree with any of the Nash equilibria of the underlying game, (b) deliberative dynamics which converged to the same deliberative outcome in The Dynamics of Rational Deliberation may lead to different deliberative outcomes here, and (c) Bayesian deliberation seems to be more likely to avoid nonstandard deliberative outcomes, contrary to the result reported in The Dynamics of Rational Deliberation, which argued in favour of the Brown–von Neumann–Nash dynamics.
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Notes
I am deliberately being ambiguous as to the nature of the phenotypic change. Most of the models used in Skyrms (1996, 2003) can be understood as models of either biological evolution or cultural evolution, and so the phenotypic change may occur as a result of strategic learning or as a result of differential rates of reproduction.
Of course, there are a variety of reasons why this type of inference might go awry. You might believe that the person you are currently interacting with is out to take advantage of you, but you believe that I will not do so. My point is that, in the absence of reasons of this kind, the inference is plausible. At least to some extent.
Informally, a connected graph has the property that, for any two vertices, there exist a path which links them. For directed graphs, where each edge (a i , a j ) is understood as “pointing” from a i to a j , one can distinguish two senses of connectedness. A strongly connected graph is a directed graph such that, for any two vertices, there exists a path which links them (where walking the path respects the directionality of the edges). A weakly connected graph is a graph where, although any two vertices are linked by a path, one may need to disregard the directionality of the edges when walking the path. Clearly all strongly connected graphs are weakly connected, but not vice versa. Since I will ultimately consider directed graphs, I must note that by “connected”, I mean “weakly connected”.
Note, though, that A i ’s state of indecision is not necessarily common knowledge amongst all of his neighbors. For example, \(A_{i_{1}}\) need not know that \(A_{i_{2}}\) knows that A i ’s state of indecision is p i (t). Common knowledge exists for the states of indecision of two players joined by an edge.
Furthermore, note that if A i were capable of conditionalising her state of indecision upon the person she was interacting with, then we would simply have a case where the pairwise dynamics investigated in The Dynamics of Rational Deliberation apply to the game played along each edge in G, modifying the conditionalised states of indecision.
Why so many? Because the states of indecision will depend upon the nature of the game one is playing. Even if you only tend to interact with a few people, say 10, on a regular basis, you will play a number of different games with each person; each of these games will likely require a different response, and so you will have a different state of indecision for each deliberative problem, even though each deliberate problem is against the same person.
Yet, having said that, I must acknowledge that the immediately following sentence in A System of Logic challenges this interpretation. Mill subsequently writes “if we knew the person thoroughly, and knew all the inducements which are acting upon him, we could foretell his conduct with as much certainty as we can predict any physical event.” The fact that Mill includes the additional qualifier “and knew all the inducements which are acting upon him” makes his claim ambiguous. Since each interaction counts as a different “inducement” acting upon a person, it would be consistent with what Mill writes to say that a person could, in principle, act differently whenever he interacts with a different person.
This does not mean that the weights will be the same across players, though, since there is an implicit dependence upon the number of edges a player is incident upon. If A i has three neighbors and A j has five, then \(w_{i i_1} = \cdots = w_{i i_3} = \frac{1}{3},\) but \(w_{j j_1} = \cdots = w_{j j_5} = \frac{1}{5}.\)
An explicit definition of both of these dynamical rules can be found in Appendix A.
No particular assumption will be made regarding what constitutes a “natural” or “reasonable” assignment of directions to edges. Quite often the same player will play as both Row and Column, although in games with different people. This represents the fact that the role we play in a game is, quite often, a matter of historical accident. Although I play as Row in an interaction with you, now, it could have been the case that I played as Column, with you playing as Row, instead. In this model, the fact that a given player permanently takes on the role of Row, in some interactions, and the role of Column, in other interactions, should be seen as a consequence of these “historical accidents” determined by social and other causal forces which lie outside the scope of the model.
See the definition of the dynamical rules in Appendix A.
The probabilities are rounded to six decimal places.
The probabilities existing in the convergent state differ here from the case discussed previously due to the higher value of the index of caution.
The remaining simulations did not converge to six decimal places within 1,000,000 iterations.
The reason for the higher value of the index of caution for Bayesian deliberation is simply due to the fact that the alternate mathematical form requires larger values to slow down the rate at which the state of indecision changes.
Provided that the index of caution is sufficiently high. If \(k<\frac{5}{4},\) Nash deliberators may arrive at one of the periodic orbits of the form identified earlier.
Assuming that the index of caution is 25.
The random graphs were generated using the following procedure: each of the 190 possible edges had a 20% chance of being included. If the resulting graph was connected, it was used; if the resulting graph was disconnected, it was thrown out and a new candidate was generated. The same graph was used for the i th trial for both Nash and Bayes deliberators, as well as the same initial conditions.
For a discussion of why these dynamics warrant the label “Bayesian”, see Skyrms (1990, pp. 36–37, 165–166).
The trivial solution 〈v 1, v 2〉 = 〈0, 0〉 has to be discarded for this reason.
Which, of course, agrees with the simulation results.
References
Brown, G., & von Neumann, J. (1950). Solutions of games by differential equations. In H. W. Kuhn, & A. W. Tucker (Eds.), Contributions to the theory of games, Vol. 24 (Annals of Mathematical Studies). Princeton, NJ: Princeton University Press.
Gigerenzer, Gerd, Todd, Peter M. & the ABC Research Group (1999). Simple heuristics that make us smart. New York: Oxford University Press.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgement under uncertainty: Heuristics and biases. New York: Cambridge University Press.
Lehrer, K., &Wagner, C. (1981). Rational consensus in science and society. Dordrecht: Reidel.
Nash, J. (1951). Non-cooperative games. Annals of Mathematics, 54(2), 286–295.
Pettit, P. (1995). The virtual reality of homo economicus. Monist, 78, 308–329.
Simon, H. A. (1957). Models of man: Social and rational. New York: Wiley.
Skyrms, B. (1990). The dynamics of rational deliberation. Cambridge, MA: Harvard University Press.
Skyrms, B. (1996). Evolution of the social contract. Cambridge: Cambridge University Press.
Skyrms, B. (2003). The stag hunt and the evolution of social structure. Cambridge: Cambridge University Press.
Acknowledgments
Many thanks to Brian Skyrms, Brad Armendt, and Kevin Zollman for their comments on an earlier draft of this paper. I would also like to thank the LSE Choice Group, who heard a talk based on the formal model used in this paper, for many useful suggestions.
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Appendix
Appendix
1.1 A. Definition of the Nash and Bayes dynamics
Let M = 〈r ij , c ij 〉| n i,j=1 be the payoff matrix for a two-player game with n strategies and let p col(t) and p row(t) be the states of indecision for Row and Column, respectively. In these states of indecision, p icol (t) and p irow (t) denote the probability assigned by Column and Row to action i in their state of indecision at time t. Thus the expected utility for action i for Row at time t is
(with the expected utility for Column defined mutatis mutandis). The expected utility of the status quo for Row at time t equals
Finally, let the covetability of act j for Row at time t be
Given this, the Nash dynamics (also known as the Brown–von Neumann–Nash dynamics) states that individuals revise their state of indecision according to the rule
where k > 0 is an index of caution specifying how quickly an agent revises his or her belief in a single iteration.
The Bayesian dynamicsFootnote 19 takes a slightly different form:
where, again, k > 0 is an index of caution. (In order to ensure that the Bayes dynamics results in a probability distribution, the payoff matrix needs to be normalized so that the lowest payoff is 0 and the greatest payoff is 1.)
1.2 B. Chicken, cycle of period 2
One can easily prove that the numerically obtained probabilities do form a cycle of period two under the dynamical rules described above, for the case where k = 1. Suppose that each player adopts the distribution v = 〈v 1, v 2〉 for the game of Chicken, where we list the strategies in the order Don’t Swerve and Swerve. We want to find values of v 1 and v 2 which, under the Nash dynamics, have the property that 〈v 1′, v 2′〉 = 〈v 2, v 1〉.
First, note that the expected utility of the status quo for both players is − 10 v 21 and the expected utility of the actions Don’t Swerve and Swerve, are, respectively, − 10 v 1 + 5 v 2 and − 5 v 1. Now, the covetability of each action is simply the difference between the expected utility of the action and the status quo, if this value is greater than zero, and zero otherwise. If we assume that the action Don’t Swerve has a nonzero covetability, then this value is − 10 v 1 + 10 v 21 + 5 v 2. If we assume that only Don’t Swerve has a nonzero covetability, then the new values of v 1 and v 2 under the Nash dynamics are
and
The system of equations given by 〈v 1′, v ′2 〉 = 〈v 2, v 1〉 has four solutions, of which only three can be probability distributions over actions.Footnote 20 The first solution is the standard mixed strategy Nash equilibrium \(\langle v_1, v_2\rangle = \left\langle \frac{1}{2},\frac{1}{2}\right\rangle;\) however, this solution must be discarded as well as it does not satisfy the assumption that Don’t Swerve has a nonzero covetability. The remaining solutions are \(\langle v_1, v_2\rangle = \left \langle \frac{1}{10} (5-\sqrt{5}), \frac{1}{10}(5+\sqrt{5})\right \rangle \) and \(\langle v_1, v_2\rangle = \left\langle \frac{1}{10}(5+\sqrt{5}), \frac{1}{10}(5 - \sqrt{5}) \right\rangle.\) Of these, only the first satisfies the assumption that Don’t Swerve has a nonzero covetability and that Swerve has a zero covetability.
All that remains is to verify that 〈v 2′, v 1′〉 = 〈v 1, v 2 〉. This can be done by straightforward calculation or by writing down a set of equations similar to the above and solving it. Note that the numeric approximations of v 1 and v 2 are 0.276393 and 0.723607, respectively, which agree with the values found by simulation.
If we consider the generalized Nash dynamics with index of caution k > 0, where a distribution is revised according to the rule
a similar analysis shows that cycles of length two exist for \(\frac{5}{4} \geq k >0.\) These cycles are given by
1.3 C. The analytic solution for the fixed-point in Chicken on a cycle of length three
Suppose that player one follows the distribution 〈1, 0 〉. We need to show that there exist a distribution 〈v 1, v 2〉 which, when adopted by players two and three, produces a stable state for all three players given the dynamics. In order for player one to remain at 〈1, 0〉, it must be the case that the notional revisions obtained for each of his pairwise interactions with two and three do not assign any probability to Swerve. If they did, then the result of averaging the two notional distributions would cause player one to move away from the distribution 〈1, 0〉.
What this means is that the covetability of Don’t Swerve and Swerve, for player one, must be zero when both player two and three adopt 〈v 1, v 2〉. Now, the expected utility of the status quo is − 10 v 1 + 5v 2 for player one, and the expected utility of the actions Don’t Swerve and Swerve are − 10v 1 + 5v 2 and − 5v 1, respectively. The covetability of Don’t Swerve is thus zero, and the covetability of Swerve will be nonzero exactly when v 1 > v 2. Requiring that player one’s distribution remains unchanged forces, then, that v 1 ≤ v 2.Footnote 21
Consider now the notional revision generated for player two in his interaction with player one. (Recall that player one is Row and player two is Column.) The expected utility of the status quo for player two is − 10v 1 − 5v 2. What are the covetabilities of Don’t Swerve and Swerve? The expected utilities of each of these actions, given that player one currently follows 〈1, 0 〉, are − 10 and − 5, respectively. From this we see that Don’t Swerve has a nonzero covetability for player two if and only if − 10 + 10v 1 + 5v 2 > 0. Given that v 1 + v 2 = 1, it is impossible for this inequality to be satisfied and so the covetability of Don’t Swerve for player two is zero.
If we assume that Swerve has a nonzero covetability for player two, then his notional revision for this particular interaction will be as follows (we use η i ab to denote the probability assigned to action i when player a calculates the notional revision for his pairwise interaction with b):
From this, it follows that η 121 < v 1 and η 221 > v 2.
Now consider what happens when player two interacts with player three. When both players use the distribution 〈v 1, v 2〉, the expected utility of the status quo for each is − 10 v 21 . What is the covetability of Don’t Swerve and Swerve for player two? Well, in order for 〈v 1, v 2〉 to be a fixed point under the dynamics, it must be the case that when player two calculates his notional revision for his interaction with player three, he finds that η 123 > v 1 and η 223 < v 2, because otherwise it would be impossible for the average of the notional revisions 〈η 121 , η 221 〉 and 〈η 123 , η 223 〉 to equal 〈v 1, v 2〉. This means that player two must find that the strategy Don’t Swerve has a nonzero covetability when he considers his interaction with player three. And, consequently, that the strategy Swerve has a covetability of zero for his interaction with player three. Hence player two will calculate his notional revision in this case as follows:
One then simply needs to solve the following three equations:
and check which solutions satisfy the assumptions made along the way. Of the four solutions, three occur in the complex plane. The remaining solution in real values is approximately 〈v 1, v 2〉 = 〈0.385492, 0.614508 〉, which agrees with the simulation results. Although of little use, it is worth seeing the complexity of the analytic solution:
and
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McKenzie Alexander, J. Local interactions and the dynamics of rational deliberation. Philos Stud 147, 103–121 (2010). https://doi.org/10.1007/s11098-009-9455-x
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DOI: https://doi.org/10.1007/s11098-009-9455-x