would lead to an uninteresting global
“chicken” strategy (all players declaring king until the final seconds of the
experiment, with some players then
“blinking” and switching to pawn),
in these experiments payoffs accrued
continuously according to the prorated time players spent in each of
the three possible states (pawn, king
with no conflicting neighbors, conflicting kings).
Every experiment was run under
two conditions—one just as described
above, and another in which the GUI
included an additional element: in
the case that the player was the lone
king in their neighborhood, and thus
enjoying the highest rate of pay, a
slider bar permitted them to specify a
fraction of their earnings in that state
to be shared equally among all their
neighbors (whose pawn status allows
the king’s high payoff). These “tips” or
side payments could range from 0% to
100% in increments of 10%, and could
be adjusted at any time. Note that in
some cases, depending on network
structure, some vertices might be able
to obtain a higher rate of pay by being
a pawn receiving side payments from
many neighboring kings than by being
the lone king in their neighborhood.
The most striking finding was that,
across a wide variety of network structures, the introduction of the side payments uniformly raised the collective
payoffs or social welfare. Side payment rates were often generous, and
averaged close to 20%. Furthermore,
when side payments are introduced,
both the average income disparity between neighboring players, and the
amount of time they spend as conflicting kings, are considerably reduced,
across all network structures examined (see Figure 6). This suggests that
without side payments, subjects used
conflict, which reduces the wealth of
all players involved, to express perceived unfairness or inequality. The
side payments reduce unfairness and
consequently reduce conflict, thus facilitating coordination and raising the
social welfare.
Biased voting. The biased voting
experiments14 shared with the earlier
consensus experiments an incentive
toward collective agreement and coordination, but with an important
strategic twist. As in consensus, each
the side payments
reduce unfairness
and consequently
reduce conflict,
thus facilitating
coordination
and raising the
social welfare.
player had to simply select a color for
their vertex, but now only between the
two colors red and blue. If within the
allotted time, the entire population
converged unanimously to either red
or blue, the experiment was halted
and every player received some payoff.
If this did not occur within the allotted time, every player received nothing
for that experiment. Thus the incentives were now not at the individual
level, but at the collective—players
had to not only agree with their neighbors, but with the entire network, even
though they were still given only local
views and interactions.
The strategic twist was that different players were paid different
amounts for convergence to the two
colors within the same experiment.
In particular, some players received a
higher payoff for convergence to blue,
while others received a higher payoff
for convergence to red. Typical incentives might pay blue-preferring players $1.50 for blue convergence and
only $0.50 for red, with red-preferring
players receiving the reverse. Some experiments permitted asymmetries between higher and lower payoffs, thus
incentivizing some players to “care”
more about the color chosen by the
population. These experiments thus
set up a deliberate tension between
competing individual preferences and
the need for collective unity.
In the most dramatic set of experiments, networks were chosen according to preferential attachment—
known to generate a small number
of vertices with high degree—and the
vast majority of players given incentives that paid more for convergence
to blue. However, the minority of vertices preferring red was chosen to be
the high-degree vertices. These experiments tested whether a small but
well-connected minority could systematically impose its preferences on the
majority, thus resulting in suboptimal
social welfare.
The answer was resoundingly affirmative: in 27 such “minority power”
experiments, 24 of them resulted in
the subjects reaching a unanimous
choice—in every case, the preferred
choice of the well-connected minority. The finding is especially surprising
when we remember that since everyone
has only local views and information,