H. Dawid J. of Economic Behavior Org. 41 2000 27–53 37
U
pr
= b √
a
1
− µ1
− r r
+ µ 1
− r r
s 1
1 + p
+ µ 1
− r r
s p
2
1 + p
= b √
a 1
− µ1
− r r
+ µ
4
√ 2
1 − r
r .
In an equilibrium the utility of traders and producers must be equal. Under the assumption that the time constraint is not binding for the traders i.e. µ
= r1 − r this yields the equation
√ 2
√ 2
− 1 r
r 1
− r =
4
√ 2.
The unique solution is r
= 1
3 √
2 − 1
= 0.8047. For χ 0.2164 we have µ
= r1 − r. The utility in this state is given by U
med
= U
pr
=
4
√ 2b
√ a.
These considerations indicate that if all agents were completely rational, completely coor- dinated and χ is sufficiently small there would be about 20 mediators in a heterogeneous
state consisting of producers and mediators. A question to be answered in the next sec- tion is whether a similar degree of organization can actually be reached by a population of
boundedly rational agents and, if it can, what kind of transient behavior can be observed. Also, the effect of the parameters in the learning rule and χ which is neglected here on
the emergence of mediation will be studied by the means of simulations.
3. Simulation results
In all our simulations we use a population of size n = 100. The population is initialized
homogeneously such that the agents decision variables are given by x
i 1
= x
i 2
= 12, s
i
= 0. The parameters a and b are both set to 1. We always assume that the agents update their
strategies every 10 periods τ = 10. The threshold for trading is given by χ = 0.1 and
the inertia parameter for the imitation process was always set to w = 2. Concerning the
innovations a normally distributed noise term with expectation 0 and variance σ = 0.1 is
added to the continuous decision variables with probability µ = 0.02. The probability that
the production focus changes from good 1 to good 2 or vice versa is in all these simulations µ
pr
= 0.05. The simulations were run for T = 3000 generations. Most of our results appear to be robust with respect to changes in these parameters. We will point out those cases where
the results are more ambiguous. Whenever we show the evolution of certain variables we present results of single simulation runs rather than average trajectories from all the runs
in a certain scenario. However, we performed sufficient numbers of runs in all scenarios to check that the results reported here are at least qualitatively robust.
38 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
The purpose of these simulations in not only to demonstrate the potential for self or- ganization a population of boundedly rational agents has — this has also been shown in
several other frameworks — but, on one hand, to study how trade emerges with time and, on the other hand, how certain features of the model like search costs or the agents’ inertia
influence the dynamics of the different variables.
3.1. Model without mediators First, we study the effect of an increase of α in the model without mediators. Our ana-
lytical considerations suggest that for small values of α the state where all agents produce both goods without trading is an equilibrium. For intermediate values of α direct trade is
profitable for the agent if it can be established but if the returns to scale increase very fast the individuals cannot gain from trade but should rather restrict themselves to the production
and consumption of a single good.
In order to explore the effects of a variation of α we increased α from 1 to 4 with a stepsize of 0.2 and performed 10 simulation runs for each value. In Fig. 1 we show the average values
of consumption, trade measured in units of good 1, degree of specialization
4
and time invested in trading after 3000 generations the superscript av always indicates that the
depicted values are average values of 10 runs. It can be seen that for α
∈ [1, 2] the population state more or less stays in the initial state where all agents produce both goods and do not trade. If we look at the trajectories of
the variables in such a setup we cannot detect any significant changes throughout the run. Due to some kind of random drift the degree of specialization and the time invested in trade
increases a little bit with time. The average utility is slightly smaller than the equilibrium value. However, there occur no relevant amounts of trade and the population remains a set
of isolated producers.
This picture changes significantly for faster increasing returns to scale. Here agents learn to specialize in the production of only one good and also the amount of time invested in
trading goes up reaching the value of χ = 0.1. The largest trade volume can be observed
for values of α close to 2.5
5
. Thus we show the trajectories of the crucial variables in a run with this parameter setting in Fig. 2 .
Even with these parameter values no significant trade emerges up to period 1500. How- ever, afterwards we can observe a sharp increase in
¯ sp, tr and ¯
U and the market very quickly changes from a regime of isolated production to one of specialized production and direct
trade. The trading volume is on average slightly below 10 units of good 1 per period. This is substantially lower than in an optimally organized market — as described in the last
section — where 19.2 units of good 1 would be traded per period. This implies that also the payoff is smaller than in equilibrium, namely ¯
U = 10.5 compared to U
tr
= 12.4. These differences are due to the fact that, contrary to our simplifying assumption that every agent
4
Since values of sp close to 0 and close to 1 both express a high degree of specialization but would give an average value of about 0.5 indicating a very low degree of specialization, we use the absolute deviation of sp from
0.5 as a measure of specialization in production.
5
Such large values of α seem to be unrealistic, however, the ‘peak’ of trade could be shifted to the left by decreasing χ . These parameter values are chosen only for matters of demonstration.
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 39
Fig. 1. a Mean value averaged over 10 runs of the average utility in the population after 3000 periods for values of α
∈ [1, 4]n = 100, a = b = 1, τ = 10, T = 3000, χ = 0.1, w = 2, σ = 0.1, µ = 0.02, µ
pr
= 0.05. b Mean value averaged over 10 runs of the amount of trade in the population after 3000 periods for values
of α ∈ [1, 4]. c Mean value averaged over 10 runs of the average degree of specialization in the population
after 3000 periods for values of α ∈ [1, 4]. d Mean value averaged over 10 runs of the average amount of time
invested for trading after 3000 periods for values of α ∈ [1, 4].
trades once every period which was used for calculating U
tr
, the agents in the simulation fail to trade in some periods.
Considering again Fig. 1 we realize that a further increase of α leads to a reduction of the amount of good 1 traded. The agents still specialize in the production of one good but
do not trade anymore. They rather keep the time invested in trading low and just consume the goods they produce. This results in average long term payoffs slightly smaller than
U
sp
= 10 which is the highest possible utility for large values of α. Note, however, that in our setup a perfectly organized economy with direct trade would give the agents a larger
40 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 1 Continued.
utility than specialization without trade for α 6.6. Fig. 1 shows that even for much smaller values of α trade cannot be established in a population of myopic individuals. This is easy
to understand because trading requires much more time in the beginning as long as there are only a few other agents who invest time in trading. Thus, the gains from trade must be
rather large to allow agents who are involved in trading to be competitive and to establish trade in the population.
Having explored the influence of the production technology on the emergence of trade we will now focus on the trading time threshold χ . In Fig. 3 we show average utility ¯
U , the trading volume tr and the trading time budget
¯s for values of χ ∈ [0.05, 0.3] and α = 2.5. It is quite obvious from this figure that the emergence of trade relies heavily on this
parameter. Values of χ larger than 0.15 prohibit the establishment of trade in the economy even with such a large value of α. As expected the degree of specialization is very high
in all these simulations which means that we either get direct trade between specialized producers, or — for large values of χ — isolated producers who produce and consume
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 41
Fig. 2. a Evolution of the average population utility in a simulation run with parameter values α
= 2.5, n = 100, a = b = 1, τ = 10, T = 3000, χ = 0.1, w = 2, σ = 0.1, µ = 0.02, µ
pr
= 0.05. b Evolution of the trading volume for the same simulation run as in Fig. 2a. c Evolution of the average degree of
specialization for the same simulation run as in Fig. 2a. d Evolution of the average time invested in trading for the same simulation run as in Fig. 2a.
42 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 2 Continued.
only one of the two goods. It is also quite remarkable how fast the amount of goods traded increases if we decrease χ from 0.1 to 0.05. For this value the average trading volume in
our simulations is even larger than in a perfectly organized economy with direct trade in such an economy we would have a trading volume of tr
≈ 22. The average utility of the agents decreases fast with increasing χ as long as this threshold is rather small but is more
or less independent from this parameter in the region above χ = 0.15 where only very little
trade occurs. It is very interesting to note that the amount of time invested in trading does not differ much between the cases where χ
= 0.05 and χ = 0.1. In both cases the value is approximately 0.1. This value seems to be a slightly larger than necessary for χ
= 0.05 and a little bit too small for χ
= 0.1 judging from the trading volumes in both cases. The value s
= 0.1 seems to be some kind of generic level of the time fraction for trading in cases where trade is established in the population. Unfortunately, we lack a sound explanation
for this phenomenon. To estimate the importance of the agents’ inertia in our model we have also carried out
simulation for different values of the parameter w which governs the degree of inertia in the learning rule. In Fig. 4 we show average utility and trade volume for w
∈ [1, 3]. Note that w
= 1 amounts to the absence of any inertia in the population. Obviously the influence of this parameter on the long run behavior of the population is
very small. We can detect a slight increase both in utility and in the amount of goods traded for increasing w but this effect is rather insignificant. A very similar picture can be obtained
if we depict utility and trade volume in dependence of the innovation probability µ
v
. As long as the probability is larger than some very small threshold and does not get too large
the level of innovation does not influence the simulation results very much. In particular, we checked that values of µ
v
∈ [0.03, 0.15] yield almost identical results. However, µ ≥ 0.02 is necessary to establish trade in the population
6
.
6
Again, we considered the case with α = 2.5 where trade is profitable for the agents.
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 43
Fig. 3. a Mean value averaged over 10 runs of the average utility in the population after 3000 periods for values of χ
∈ [0.05, 0.3]α = 2.5, n = 100, a = b = 1, τ = 10, T = 3000, w = 2, σ = 0.1, µ = 0.02, µ
pr
= 0.05. b Mean value averaged over 10 runs of the trading volume in the population after 3000 periods for values of
χ ∈ [0.05, 0.3]. c Mean value averaged over 10 runs of the average amount of time invested in trading after
3000 periods for values of χ ∈ [0.05, 0.3].
3.2. Model with mediators Now let us consider the model where agents might decide to mediate rather than pro-
duce. The parameter settings are the same as in the simulations presented above and, ad- ditionally, we specified the probability to change from mediation to production or vice
versa as µ
id
= 0.01, the probability that the selling and buying prices of a mediator are perturbed by noise is µ
= 0.02 and the variance of the normally distributed noise is σ
= 0.1.
44 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 3 Continued.
Again, we start by considering the effects of a change of the parameter α on the outcome of an average simulation. In Fig. 5 we show the average utility of the agents, the amount of
trade and the number of mediators in the population for α ∈ [1, 4].
Looking at this figure we get the following picture: for small α — like in the model without mediation — no trade emerges but the agents keep producing and consuming
both goods. If α is larger than 1.6 trade occurs in the population and it follows from the number of mediators and the average time non-mediators invest in trading that this trade is
mediated. However, for α ∈ [1.6, 2.2] mediators earn on average less than the producers and
thus the number of mediators and accordingly the trade volume is rather small. Nevertheless, the proportional imitation rule of the agents allows a small number of mediators to survive.
Most probably this would not be the case if we would use a stronger rule like the ’imitate the best’ rule. If α 2.2 the long run outcome of the simulations very much looks like the
equilibrium with mediation we considered in the last section. We have a large degree of specialization of the producers and they do not invest in trading. There are between 10 and
14 mediators in the market and the utility of producers and mediators is almost equal. Even the amount of goods traded is very close to the 24 units of good 1 which would be traded
in an equilibrium with mediation. The utility of the agents is almost constant for α 2.2 and significantly larger than in the model without mediation.
Fig. 5 suggests that, if we allow the agents to become middlemen in trade, in the long run we always get mediated trade if we get trade established at all. An interesting question
in this context is whether the evolution of mediated trade needs a stage with direct trade as step in-between or whether mediation emerges directly from a population of pure producers.
To answer these questions we have to look at the time evolution of the population in the model with mediators. A typical simulation result for α
= 2.5 is depicted in Fig. 6 . The population evolves towards a state which is very similar to an equilibrium with mediation.
The average utility in the equilibrium with mediation is U
med
= 11.89 note that this value is independent from the value of α. Fig. 6a shows that the average payoff in the population
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 45
Fig. 4. a Mean value averaged over 10 runs of the average utility in the population after 3000 periods for values of w
∈ [1, 3]α = 2.5, n = 100, a = b = 1, τ = 10, T = 3000, χ = 0.1, σ = 0.1, µ = 0.02, µ
pr
= 0.05. b Mean value averaged over 10 runs of the trading volume after 3000 periods for values of w
∈ [1, 3].
oscillates close to this value in our simulation. Furthermore, we get strong specialization of all agents and between 10 and 15 mediators in the population facilitating trade. Interestingly
enough, the average price charged by the mediators is significantly smaller than the optimal price would be. This inefficiency seems to be the reason why less mediators than predicted
are in the market and the overall utility in the population is smaller than the value calculated analytically.
Fig. 6f shows the average payoff of a producer compared to the average payoff of a me- diator. It can be clearly seen that the amplitude of the oscillations of the mediators’ payoff
is much larger than that of the producer. This is quite intuitive, because the number of mediators has a much stronger influence on the payoff of a mediator than on the payoff of a
producer and — as we can see in Fig. 6d — the number of mediators shows substantial
46 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 5. a Mean value averaged over 10 runs of the average utility in the population after 3000 periods in the model with mediators and values of α
∈ [1, 4]n = 100, a = b = 1, τ = 10, T = 3000, χ = 0.1, w = 2, σ
= 0.1, µ = 0.02, µ
pr
= 0.05, µ
id
= 0.01. b Mean value averaged over 10 runs of the trading volume after 3000 periods in the model with mediators and values of α
∈ [1, 4]. c Mean value averaged over 10 runs of the number of mediators in the population after 3000 periods for values of α
∈ [1, 4]. d Mean values averaged over 10 runs of the average utility of mediators dotted line and producers solid line after 3000 periods for values
of α ∈ [1, 4].
oscillations. However, the average payoffs of producing and mediating is very close con- sidering the average payoff between period 1000 and 3000 we get 11.34 for the producers
and 11.1 for the mediators. Considering the question whether direct trade evolves and is afterwards replaced by me-
diated trade we realize that no direct trade can be observed at any stage of the simulation. This is in accordance with our observations throughout these simulations. In all our simula-
tions with different parameter constellations in this model we never observed a simulation
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 47
Fig. 5 Continued.
of the model with mediation where the population in the long run traded via mediators but developed direct trade before
7
. Using Fig. 3, above we analyzed the effect of an increase of the trading time threshold
χ on the emergence of trade in the model without mediators. How does this effect change if mediation is an option for the agents? In order to analyze this question we present the
average utility, the trade volume and the number of mediators for χ ∈ [0.05, 0.3].
Comparing Fig. 7 with Fig. 3 we see that the effect of an increase of χ on the average utility is smaller in the model with mediators. Utility still decreases with increasing χ but
7
Contrary to this we demonstrate in Dawid 1999 in a similar model with decreasing returns to scale in production where different agents have different production technologies, that a sequence of all three modes of behavior no
trade, direct trade and mediated trade can be observed in simulation runs.
48 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 6. a Evolution of the average population utility in the model with mediators and parameter values α
= 2.5, n = 100, a = b = 1, τ = 10, T = 3000, χ = 0.1, w = 2, σ = 0.1, µ = 0.02, µ
pr
= 0.05, µ
id
= 0.01. b Evolution of the trading volume for the same simulation as in Fig. 6a. c Evolution of the number of mediators
for the same simulation as in Fig. 6a. d Evolution of the average degree of specialization in the population for the same simulation as in Fig. 6a. e Evolution of the average buying and selling prices of mediators for the same
simulation as in Fig. 6a. f Evolution of the average utility of mediators solid line and producers thick line for the same simulation as in Fig. 6a.
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 49
Fig. 6 Continued.
50 H. Dawid J. of Economic Behavior Org. 41 2000 27–53
Fig. 7. a Mean value averaged over 10 runs of the average utility in the population after 3000 periods in the model with mediators and values of χ
∈ [0.05, 0.3]α = 2.5, n = 100, a = b = 1, τ = 10, T = 3000, w
= 2, σ = 0.1, µ = 0.02, µ
pr
= 0.05. b Mean value averaged over 10 runs of the trading volume after 3000 periods in the model with mediators and values of χ
∈ [0.05, 0.3]. c Mean value averaged over 10 runs of the number of mediators after 3000 periods for χ
∈ [0.05, 0.3]. d Mean value averaged over 10 runs of the average time invested in trading after 3000 periods in the model with mediators and values of χ
∈ [0.05, 0.3].
much slower than in Fig. 3a. The same holds true for the amount of trade, which virtually vanishes for χ 0.15 in the absence of mediation but still emerges for χ
= 0.3 if the agents have the option to start mediating. It is also interesting to see that the number of
mediators which are on average in the long run present in the population are more or less constant over a wide range of χ . This implies that for these values of χ the time constraint
is not binding for the mediators. For a very small threshold χ the number of mediators is comparably small and
¯s is rather large. This indicates that for such a low trading time threshold in some runs a certain part of the trade is carried out directly. We checked this
H. Dawid J. of Economic Behavior Org. 41 2000 27–53 51
Fig. 7 Continued.
conjecture and could indeed observe such outcomes in different simulation runs with this parameter constellation, however not in all of them.
4. Conclusions
