On the coevolution of social norms in primitive societies

This literature is concerned with evolutionary dynamics in scenarios characterized by investment specificity. We say that investment is completely relation-specific when it is only valuable to a particular trading partner. By making investor vulnerable to ex-post exploitation, investment specificity may thus give rise to the so-called hold-up problem. Relation-specific investment seems a minor problem for nomadic hunter–gatherers since, for their subsistence, they are not dependent on specific other agents, nor on specific assets and resources; see Woodburn (1982).

Other important contributions include Young (1993a) and Hawkes (1992). The former explains the evolution of a division norm when the pie is exogenously given; the latter explains the evolution of a norm regulating agents’ cooperation in producing a pie, when the division rule is exogenously given.

More precisely, they show that the evolved division norm virtually assigns the entire surplus to the unique investing agent, provided that a fine grid of investment choices is allowed.

Dawid and MacLeod (2008) is a further extension in which the outcome of the investment decision is stochastic.

This is not the evolutionary framework considered by Dawid and MacLeod (2001) who are basically concerned with an adaptation of Young (1993b) to an extensive form game. Although this extension is not problematic with one-sided investment (as in Troger 2002), it is a bit tricky with two-sided investment since it can imply that some agents continue to believe that all the opponents make the same investment (i.e. all choose high or low investments) even when some bargaining outcomes (which in their model can only happen when high-low matches occur) are observed.

In a preliminary study we considered the case in which a NDG, rather then an UG, occurs after an asymmetric investment profile; however we were not able to derive the stochastically stable equilibrium due to the occurrence of several non-singleton absorbing sets with complex structure. Ellingsen and Robles (2002) also considered the case in which, in stage two, the distribution of the surplus is determined by an ultimatum game where the player who makes the proposal is the trading partner, i.e. the agent not responsible for the investment decision. They have shown that in this case the stochastic stability has little cutting power because many outcomes are stochastically stable. Our game \(\varGamma _{UG}\) mainly differs from Ellingsen and Robles (2002) in two respects. First, the player who makes the proposal is the player who decided to invest in the first stage. Second, since the pie depends on the decisions of two agents, both can be in a position to affect the distribution of the surplus generated by the other. Our result for \(\varGamma _{UG}\) says that, under the appropriate conditions, a unique stochastically stable outcome exists.

Woodburn (1982) suggests that among the Hazda and the !Kung this “right to a share” by those not investing is responsible for the overwhelming difficulties encountered by those who make an effort to live by agriculture.

The connection between complementarity and equal sharing is also suggested, albeit implicitly, by Boehm (2012). He maintains that team hunting is an energetically demanding occupation which works best if all the members of the hunting team are adequately nourished, a condition allowed by equal sharing.

It is worth observing that, although the formal conditions for a stochastically stable outcome to exist in \(\varGamma _{UG}\) coincide with those required by Dawid and MacLeod (2001), the basic models and the evolutionary dynamics are different. In particular, while in \(\varGamma _{UG}\) these conditions support the coevolution of a norm of cooperation and a norm of division, they only uphold a norm of investment in Dawid and MacLeod (2001).

In \(\varGamma _{UG}\), a plan of action for player A must specify: (i) the type of investment; (ii) the demand when both players choose H (the action at HH); (iii) the demand when A chooses H and B chooses L (the action at HL); (iv) whether to accept or reject any demands made by B, when in the first stage B chooses H and A chooses L. The same applies for player B. In \(\varGamma _{DG}\), a plan of action for player A must specify: (i) the type of investment; (ii) the demand when both players choose H (the action at HH); (iii) the division of the surplus when A chooses H and B chooses L (the action at HL). Analogously for player B.

However, if the learning agent has already played a best reply her action does not change. Moreover, when the best reply contains more than one action, one of these can be randomly chosen according to a distribution with full support.

A set \(\varOmega \subseteq \varTheta \) is called a limit set of the process \( \left( \varTheta ,P\right) \) if: (a) \(\forall \theta \in \varOmega \), \(Prob\left\{ \theta _{t+1}\in \varOmega \mid \theta _{t}=\theta \right\} =1;\) (b) \(\forall \left( \theta ,\theta ^{\prime }\right) \in \varOmega ^{2}\), \(\exists s>0\) s.t. \(Prob\left\{ \theta _{t+s}=\theta ^{\prime }\mid \theta _{t}=\theta \right\} >0.\)

According to Noldeke and Samuelson (1993) a state is a self-confirming equilibrium if each agent’s strategy is a best response to that agent’s conjecture and if each agent’s conjecture about opponent’s strategies matches the opponent’s choices at information sets that are reached in the play of some matches.

\( {\widehat{x}}_{A}^{U}\) is the largest demand agent B can make at HH such that A does not have any incentive to change action by playing L when she knows that: (i) \(N-1\) agents B play H and claim \({\widehat{x}}_{A}^{U}\); (ii) one agent B makes a larger demand. Analogously for \({\widehat{x}} _{B}^{U}\).

Our results can be compared with Dawid and MacLeod (2001). If we put the assumptions \(V_{L}=0\) and \(V_{M}-c>0\) into their model, then the formal conditions for a single stochastically stable outcome stated in their Proposition 7 are in line with those stated in our Proposition 3. However their Proposition 7 is only concerned with the evolution of investment norms instead of the coevolution of investment and bargaining norms. As we said, this stems from deep differences between the two models and the evolutionary dynamics considered. In the preliminary version of the present paper we have also studied the model in which the surplus is equally split when both agents invest, as in Dawid and MacLeod (2001), but an Ultimatum Game occurs when only one agent has invested. In this case Proposition 3 continues to be true. Lastly, Proposition 3 continues to hold even when in the UG the agent who makes a proposal is not the agent who has chosen to invest, as in Ellingsen and Robles (2002).

We remind that in region NN condition (5) does not hold.

See Heinrich et al. (2004). An assessment of these experiments can be found in Chibnik (2005) and in Hagen and Hammerstein (2006).

This game tallies with Hawkes’s game under the assumptions \(V=V_{H}\) and \( sV=V_{M}.\)

This means that the distributional rule in \(\varGamma _{C}\) coincides with the unique distributional norm which can evolve in \(\varGamma _{UG}.\)

According to the anthropologists, this situation is compatible with societies admitting the so-called tolerated theft (Hawkes 1992). This means that sharing also occurs when the pie is only provided by one agent, since excluding outsiders is too costly. However, Bell (1995) argued that tolerated theft presumes that society ensures the hunter the full right to his or her catch, a condition that may not be granted.

We remark that in \(\varGamma _{C}^{\prime }\) any distributional norms granting a positive payoff when both invest does not hamper the evolution of the investment norm.