José Castro Caldas and Helder Coelho (1999)
Journal of Artificial Societies and Social Simulation vol. 2, no. 2, <http://jasss.soc.surrey.ac.uk/2/2/1.html>
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Received: 2-Feb-99 Accepted: 10-Mar-99 Published: 14-Apr-99
As it seems to me, the feelings of which a man is capable are of various grades. He is always subject to mere physical pleasure or pain [...]. He is capable also of mental and moral feelings of several degrees of elevation. A higher motive may rightly overbalance all considerations belonging even to the next lower range of feelings; but so long as the higher motive does not intervene, it is surely both desirable and right that the lower motives should be balanced against each other[...]. Motives and feelings are certainly of the same kind to the extent that we are able to weigh them against each other; but they are, nevertheless, almost incomparable in power and authority.
My present purpose is accomplished in pointing out this hierarchy of feeling, and assigning a proper place to the pleasures and pains with which the Economist deals. It is the lowest rank of feeling which we here treat. [...] Each labourer, in the absence of other motives, is supposed to devote his energy to the accumulation of wealth. A higher calculus of moral right and wrong would be needed to show how he may best employ that wealth for the good of others as well as himself. But when that higher calculus gives no prohibition, we need the lower calculus to gain us the utmost good in matters of moral indifference.
randomly generate a population P(t)
determine aggregate results for P(t)
while not(stopping criteria) do
for every agent j do
assign credit to rule sets in P(t)
select a rule set aj1 from P(t)
if r < prob_crossj then
select a rule set aj2 from P(t)
crossover(aj1, aj2) to a'j
for every bit in a'j
if r <prob_mutj then mutate that bit
determine aggregate results for P(t)
t <- t+1
|Figure 1: Co-ordination (problem 1): Choice frequency per colour through the simulation|
|Figure 2: Co-ordination (problem 2): Choice frequency per colour through the simulation|
we observe, as expected, a process of convergence. In the run reported in figure 2, choices that are Pareto inefficient (colours b and e) compete at the beginning of the simulation, and colour b that was not the most frequent in the initial population finally defeats colour e. Five of ten runs of the simulation with different initial populations converged to the Pareto optimal choice, the remaining simulations converged to a Pareto inferior outcome. Once again an invisible hand guides the agents, except that now it may lead them to an outcome that is not the best possible. We may conjecture, however, that if we would let the players discuss and agree on a joint strategy the chances of co-ordination in a Pareto optimal choice would increase.
There are many theories. One, the economic/game-theoretical prediction, is that no one will ever contribute anything. Each potential contributor will try to "free-ride" on the others. [...] Another theory, which I will call the sociological-psychological prediction, is that each subject will contribute something [...] it is some times claimed that altruism, social norms or group identification will lead each to contribute [...x...], the group optimal outcome. [...] Examination of the data reveals that neither theory is right.As a matter of fact, the experimental evidence in similar cases shows that, with large groups, positive posted contributions are observable in the first rounds but free-riding soon emerges leading the group to levels of contribution that all agents consider undesirable.
The collective return on investment is given by:
The apportioning rule is:
The credit assignment function is:
and the collective payoff is given by:
|Figure 3: probability of monitoring = 0. Contributions and collective payoffs through the simulation|
|Figure 4: probability of monitoring = 1. Contributions and collective payoffs through the simulation|
with the implicit penalty reverting to the meta-agent and included in the collective payoff. The meta-agent chooses the individuals to be inspected by a simple rule: if r (a random real between 0 and 1) is lower than probability of monitoring (a parameter of the simulation), then agent i's envelope will be opened.
For non-monitored actions, under both regimes, the credit of action i to agent j is assigned by:
and for monitored actions with contributioni < 35 we have
The collective payoff is given by (A5), and the size of one agent is updated according to:
The simulation includes a training period of 100 generations during which no voting takes place and which is used by the agents to explore the regimes of the two rules, assigning values to them. Rule 2 is experienced in the initial fifty generations and rule 1 throughout the next fifty. In generation 101, and every 20 generations after that, a vote takes place that may change the rule regime.
|Figure 5: all agents created equal: contributions and collective payoffs through the simulation|
|Figure 6: distributed power: contributions and collective payoffs through the simulation|
|Figure 7: distributed power: number of votes and voters|
2 Arifovic's use of the GA has been discussed by Chattoe (1998) and in Caldas and Coelho (1999).
3 P(t) stands for the population in generation t, r is a uniformly distributed random number between 0 and 1; prob_crossj and prob_mutj are the parameters that set the probability of crossover of a selected chromosome and the probability of mutation of each single bit for agent j.
4 Even though in this particular context mutation is hardly justifiable in rational terms, note that instability in convergence is also observable with human beings in similar experimental contexts; boredom or insufficient understanding of the game situation are usually the explanations given by researchers. The authors thank an anonymous referee for pointing out that an important feature, that has been experimentally observed is not captured by the model: some strategies, for instance 'black' and 'white' strategies, may be prominent. In multiple runs in the laboratory they would be observed as outcomes with a non-random frequency.
5 In all the simulations that follow, the Population Size is 20, the Probability of Crossover is 0.5 and the Probability of Mutation is 0.01, for all agents.
6 Different initial populations were generated using various random generator seeds. The observed overall pattern of outcome is common to all.
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