View source for Hawk Dove Game and the Evolution of Property Rights

== The First Property Rights Revolution ==
'''Background'''

In their paper, The First Property Rights Revolution, Samuel Bowles and Jung-Kyoo Choi attempt to explain the emergence of individual property rights, which replaced the "collectivist and egalitarian social structures" common to mobile foraging bands.  A key element in this investigation is why property rights only emerged about 11,000 years ago with the domestication of plants and animals, and how they were able to evolve without the aid of states which did not appear until many millennia later.  From archeological evidence and recent behavioral experiments it is clear that sharing norms existed as early as 100,000 years ago, and that the lives of these mobile hunter-gatherers were regulated by social norms and collective punishment of violators of these norms.  These norms most likely applied to certain items, like large game, which were available sporadically and in large quantities.

"In these ... communities, group sanction emerged as the most powerful instrument for regulation of individually assertive behaviors, particularly those which obviously disrupted cooperation or disturbed social equilibrium needed for group stability."

In addition, humans are not unique in having property rights.  Many species of spiders and the male Hamadryas baboons respect prior possession of sites and objects by individuals of the same species.  These facts are puzzling in the face of explanations which present property rights as a way to minimize the waste from conflict, but cannot explain why property rights did not develop nearly 100,000 years earlier.

Approximately 11,000 years ago there was a significant climate change which allowed the domestication of plants and animals to become a feasible strategy.  The process of transition to agriculture was gradual, but eventually replaced the social orders of all but a few hunter-gatherer bands.  The connection between domestication and the emergence of property rights is that the possession of land and livestock is relatively straight forward.  There is likely to be little confusion as to whose land and animals are whose, versus a large animal that many people work together in different manners to kill.  It also appears that initially farming reduced the productivity of labor, but increased the productivity of land, making smaller richer patches of land more valuable and deterring a more mobile living style.  This may be why successful agriculture did not predate the emergence of property rights.  According to two leading archeologists, Robert Braidwood and Gordon Willey, agriculture did not thrive where property rights did not exist because "culture was not ready".


'''The Model'''

The model proposed by Bowles and Choi is similar to the Hawk-Dove-Bourgeois Game that we have seen in class.  In the model there are n members of a foraging band who divide a good of value v.  Individuals may adopt one of three strategies: grabbing (Hawk), sharing (Dove), or punishing.  When a Grabber meets a sharer they take the good.  When a Grabber meets a Grabber they fight. They either gain the good or bear the cost of defeat, c>v, with equal probability.  When a Punisher meets a Grabber, all the Punishers try to punish the Grabber.  If they win, the good is distributed amongst all Punishers, or bear the cost of defeat c.  The probability of successfully punishing a Grabber is equal to the population frequency of punisher, ?.  If ? is the population frequency of Sharers, the payoffs are as follows:

[[Image:Bowlespayoffs.JPG|thumb|Payoffs in the punishment game]]

with expected payoffs for each strategy:

?s=(?+?)1/2v

?p=(?+?)1/2v+(1-?-?)(?v-(1-?)c)

?g=?v+?{(1-?)v-?c}+(1-?-?)1/2(v-c)

The cultural transmission process is based on the above payoffs according to a payoff monotonic updating process, where n is large enough that actual payoffs is approximate to expected payoffs.  Thus:

d?/dt=?(?s-?av)

d?/dt=?(?p-?av)

where average payoff, ?av, is

?av=??s+??p+(1-?-?)?g

[[Image:Bowlesgroupdynamic3.JPG|thumb|Within-group dynamics]]

The vectors indicate the direction of movement in the regions defined by the loci along which ?,?, and ? are stationary.  For this figure v/c=2/3.

Two of the stationary outcomes of this model are particularly interesting.  Point b corresponds to the classic equilibrium found in the Hawk Dove game, and is asymptotically stable (cannot be invaded by punishers).  The left side of the triangle corresponds to an all Sharer-Punisher composition.  These points are stable (not invadable by grabbers), but not self-correcting (subject to drift).  This second equilibrium appears to be what most of human history saw in terms of social arrangements, however, allowing for slight disruption (chance events) this equilibrium will not persist over time.  In this equilibrium Sharers and Punishers receive the same payoffs, so eventually Sharers will make up a large enough part of the population that Grabbers will be able to invade, falling back to the traditional Hawk Dove equilibrium.  The Hawk-Dove does not appear to be what happened and several reasons could explain why the Sharer-Punisher equilibrium would be stable.  If groups periodically encounter adversity from the environment or conflict with other groups, the groups with higher average returns have a better chance of survival.  In addition, if there is conformist cultural transmission, this will work against drift and allow all Punishers to be a stable equilibrium, since when there are no Grabbers, Sharers and Punishers have the same payoffs.  There might also have existed a second-order punishment which in addition to punishing Grabbers, punished Sharers as well, who free ride on the collective punishment of Grabbers.  The all Punisher equilibrium would be stable even if the punisher incurred a cost and the cost to the Sharers of the second-order punishment was relatively small (perhaps temporary shunning, or slightly less of a shared resource), since the punishment would only be necessary to prevent drift.

[[Image:Bowlesavpayoff.JPG|thumb|Average payoffs and between-group dynamics]]

The contour lines show distributions of the three strategies  with the same group level average payoff.

Second-order punishment, conformist cultural transmission, and adversity which gives group members a shared fate presents a plausible case for the Punisher equilibrium persisting over long periods of time.  The Bourgeois strategy in the Hawk Dove game is an evolutionarily stable strategy and can invade the traditional Sharer-Graber equilibrium mentioned above.  This will create a new asymptotically stable equilibrium in which there are no Punishers, Grabbers or Sharers with average payoff being the same as the Punisher-Sharer equilibrium (v/2).  It seems puzzling that this strategy would not be implemented until the development of agriculture if it is such a good strategy.  A possible explanation that Bowles gives is that the development of agriculture made possession a more definite concept.  Ownership of large foraging territory would have been very difficult to define and enforce.  With definite property rights individuals can use the concept of ownership to eliminate conflict.  This is modeled by having Bourgeois players "mistakenly" play as a Grabber some fraction of the time ?.  The Bourgeois equilibrium exists then only if it is a mutual best response when played against itself.

1/2[(1-?)v+?1/2(v-c)]+1/2?1/2(v-c)=1/2(v-?c)

The first term describes the probability with which the individual is a possessor and encounters a Sharer ("correct" Bourgeois) (1-?) of the time, and ? of the time is playing against a Grabber ("mistaken" Bourgeois) with the conflict payoff 1/2(v-c).  The second term is a repeat of this interaction, where the individual is the intruder.

The expected payoff to a Grabber in an almost completely Bourgeois population is:

1/2v(1-?)+1/4(v-c)(1+?) for ?<1

This is clearly less than the payoffs to Bourgeois played against itself, so Grabbers will not be able to invade a Bourgeois population, where property rights are anything but completely ambiguous (which would mean there would be no distinction between Bourgeois and Grabber anyway).  The payoffs to Sharers (1-?)v/4 however, is a best response to some values of ?<1.  Therefore a foraging way of life, with ambiguous property rights will not support a Bourgeois equilibrium, however, a domesticated way of life that makes property rights relatively unambiguous would allow a Bourgeois equilibrium to persist over long periods of time.

The next step is to model cultural transmission.  In the model the number of group members remain the same throughout, and the passage of generations is marked by a time period when members adopt new behaviors.  Once each generation, each member is paired with a cultural model.  If the member and the model are the same type then the member keeps his trait.  If the model is of a different type than the member, the member compares his payoffs this period to the model's payoffs and switches if the model's payoffs were higher.  The pairing rule allows conformism into the picture by making the chance of being selected as a model change with the proportion of that group.  In the case of a Sharer, the probability that they will be drawn as a cultural model is:

?^n/(?^n+?^n+?^n)

The probability of a Grabber or Punisher being drawn as a cultural model is determined in a similar fashion.  n is a measure of the biased cultural transmission.  For n=1 the pairing is random and for n>1 larger groups contribute proportionally more cultural models.

[[Image:Bowlestraitfreq.JPG|thumb|Description]]

In addition, each generation the group engages in conflict with a random neighbor (According to Bowles, Choi, and Hopfensitz (2002) warfare was probably much more common than this).  The conflict is modeled in such a way that the group with the higher average payoff wins the conflict and the losing group's members are assimilated into the winning group  The cultural models for the losing group are drawn from the winning group with n>1, since the winners help socialize the next generation of losers.

[[Image:Bowlessimulation.JPG|thumb|Description]]

So how does this affect the introduction of Bourgeois players into the game?  With the introduction of agriculture making property rights feasible we can replace the Grabbers with Bourgeois because we know that Bourgeois will act like grabbers when property rights are mistaken, but will receive higher payoffs than Grabbers as long as property rights are ever correctly identified.  Therefore they will be eliminated by the modeled evolutionary forces.

[[Image:Bowlesproprights.JPG|thumb|Description]]

Each point represents the average frequency of 20 simulations of 10,000 generations.

It is clear from this model that the Sharer-Punisher equilibrium is sustainable, and can resist Grabber or Bourgeois invasion as long as conformist transmission, second-order punishment, and ambiguous property rights are in effect.  With more certain property rights, however, this equilibrium cannot be maintained.  This model supports the historical relevance of the Sharer-Punisher equilibrium and the essential role of agriculture in the certainty of property rights.  This model also shows that the domestication of plants and animals surely did not precede the development of property rights [[Sources| [31]]].



<p align="center"> [[General Overview]] | [[Major Contributors]] | [[Game Theory Models]] | [[Objections/Arguments]] | [[Sources]]</p>