network structure manifests itself visually in the tendency for these groups
of six to change colors approximately
simultaneously. As was typical, after
an initial diversity of colors, the population quickly settles down to just two
or three, and nearly converges to blue
before a trickle of orange propagates
through the network and takes firm
hold; at some point the majority is orange, but this wanes again until the
experiment ends in deadlock. Acts of
individual signaling (such as toggling
between colors) and (apparent) irrationality or experimentation (playing a
color not present anywhere else in the
network) can also be observed.
Networked trading and bargaining. Our experiments on trading and
bargaining differ from the others in
that they are accompanied by nontrivial equilibrium theories that generalize certain classical microeconomic
models to the networked setting.
4, 11 In
the networked trading experiments,
4
there were two virtual goods available
for trade—call them milk and wheat—
and two types of players: those that
start with an endowment of milk, but
whose payoff is proportional only to
how much wheat they obtain via trade;
and those that start with wheat but
only value milk. All networks were bipartite between the two types of players, and trade was permitted only with
network neighbors; players endowed
with milk could only trade for wheat
and vice-versa, so there were no “
resale” or arbitrage opportunities. All
endowments were fully divisible and
equal, so the only asymmetries are
due to network position. The system
GUI allowed players to broadcast to
their neighbors a proposed rate of exchangec of their endowment good for
the other good in the form of a traditional limit order in financial markets,
and to see the counter offers made by
their neighbors; any time the rates of
two neighboring limit orders crossed,
an irrevocable trade was booked for
both parties.
For the one-shot, simultaneous
trade version of this model, there is
a detailed equilibrium theory that
c As per the theoretical model, players were not
able to offer different rates to different neighbors; thus conceptually prices label vertices,
not edges.
precisely predicts the wealth of every
player based on their position in the
network;
11 in brief, the richest and
poorest players at equilibrium are
determined by finding the subset of
vertices whose neighbor set yields the
greatest contraction,d and this can
be applied recursively to compute all
equilibrium wealths. An implication
is that the only bipartite networks
in which there will not be variation
in player wealths at equilibrium are
those that contain perfect matchings. One of the primary goals of the
experiment was to test this equilibrium theory behaviorally, particularly
because equilibrium wealths are not
determined by local structure alone,
and thus might be challenging for human subjects to discover from only local interactions; even the best known
centralized algorithm for computing
equilibrium uses linear programming
as a subroutine.
5 We again examined
a wide variety of network structures,
including several where equilibrium
predictions have considerable variation in player wealth.
There were a number of notable
findings regarding the comparison of
subject behavior to the equilibrium
theory. In particular, across all experiments and networks, there was strong
negative correlation between the equilibrium predicted variation of wealth
across players, and the collective earnings of the human subjects—even
though there was strong positive
correlation between equilibrium wealth
variation and behavioral wealth variation. In other words, the greater the
variation of wealth predicted by equilibrium, the greater the actual variation in behavioral wealth, but the more
money that was left on the table by the
subjects. This apparent distaste for
unequal allocation of payoffs was confirmed by our best-fit model for player
payoffs, which turned out to be a mixture of the equilibrium wealth distribution and the uniform distribution
in approximately a (3/4; 1/4) weighting. Thus the equilibrium theory is
definitely relevant, but is improved by
tilting it toward greater equality. This
d For instance, a set of 10 milk players who collectively have only three neighboring wheat
players on the other side of the bipartite network has a contraction of 10/3.
can be viewed as a networked instance
of inequality aversion, a bias that has
been noted repeatedly in the behavioral game theory literature.
1
Our experiments on networked
bargaining3 have a similarly financial
flavor, and are also accompanied by
an equilibrium theory.
4 In these experiments, each edge in the network represents a separate instance of Nash’s
bargaining game:
21 if by the end of
the experiment, the two subjects on
each end of an edge can agree on how
to split $2, they each receive their negotiated share (otherwise they receive
nothing for this edge). Subjects were
thus simultaneously bargaining independently with multiple neighbors
for multiple payoffs. Network effects
can arise due to the fact that different
players have different degrees and thus
varying numbers of deals, thus affecting their “outside options” regarding
any particular deal. In many experiments, the system also enforced limits
on the number of deals a player could
close; these limits were less than the
player’s degree, incentivizing subjects
to shop around for the best deals in
their neighborhood. The system provided a GUI that let players make and
see separate counter offers with each
of their neighbors.
Perhaps the most interesting finding regarded the comparison between
subject performance and a simple
Figure 5. human performance vs. greedy
algorithm in networked bargaining,
demonstrating the effects of subject
obstinacy. Where occlusions occur,
blue dots are slightly enlarged for visual
clarity. the length of the vertical lines
measure the significant effects of subject
obstinacy on payoffs.
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