Communications - October 2012

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.