Mixed-Strategy Kant-Nash Equilibrium and Private Contributions to a Public Good

Laffont (1975) was the first paper to model Kantian behavior in the context of public goods. Cornes and Sandler (1984) made a brief reference to Kantian ethics: “…the Pareto path corresponds to Kantian behaviour, since the ‘categorical imperative, whereby each acts as they want others to act, is satisfied” (p. 377). I thank Wolfgang Buchholz for this quotation.

For example, the level of cleanliness of the sidewalks in a typical suburb in Kobe, Japan, is much higher than that of a typical suburb of Montreal, Canada with the same per capita income. On a positive note, Montreal’s student ghetto near McGill University has gradually become cleaner over the past two decades, suggesting that the Kantian mode of behavior has become more widespread.

See Smith (1790, [2002]), Part III, Chapter V, p. 190. Smith’s view was echoed in Leif Johansen (1976), who wrote that economic theory “tends to suggest that people are honest only to the extent that they have economic incentives for being so”, and went on to argue that “the assumption can hardly be true in its most extreme form. No society would be viable without some norms and rules of conduct.”

Roemer (2015) also introduces the ‘Kantian variation’ which contains additive Kantian equilibrium and multiplicative Kantian equilibrium as special cases. The Kantian variation is reminiscent of the notion of conjectural variation, which was insightfully used by Cornes and Sandler (1984) in the analysis of non-Nash behavior in public good games. See also Buchholz et al. (2014a) and Buchholz (2016).

Roemer (2010) has an example of a two-person prisoner dilemma game in which mixed strategies are allowed.

The concept of Kant-Nash equilibrium in pure strategies was developed in Long (2016), and has been generalized by Grafton et al. (2016).

In a model with only two Kantians and no Nashian, Roemer (2010) also showed the possibility of a mixed strategy equilibrium. He wrote that “only if utility from cheating is not too high will Kantian ethics induce full cooperation.” That statement is somewhat misleading, as it incorrectly equates “full cooperation” with “always contributing.” It should be restated as follows: The strategy profile “always contributing” (i.e. choosing C with probability 1) is a collusive equilibrium if and only if a player’s personal gain from non-contributing (while others contribute) is sufficiently low”. In fact, when a player’s personal gain from non-contributing (while others contribute) is high enough, all players will agree that they will be better off by choosing a common non-degenerate mixed strategy, and a social planner would advise them to do the same thing.

See Bertrand Russell’s A History of Western Philosophy. As explained by Russell (1945, p. 737) ‘There are two sorts of imperative: the hypothetical imperative, which says “You must do so-and-so if you wish to achieve such-and-such an end”; and the categorical imperative, which says that a certain kind of action is objectively necessary, without regard to any end.’

For examples of replicator dynamics, see Bala and Long (2005), Breton et al. (2010), Sethi and Somanathan (1996). For an account of the evolution of cooperation in human society, see Seabright (2010).