Several network game experiments have been conducted. Most experiments are based on the BG model; only few follow a similar specification as the JW model, which is the focus of the RL model analysed later. However, also for the (partial) cooperative JW model some conclusions can be drawn from the experimental literature.
Vanin(2002) conducts an exploratory experiment of the JW model with four players. The cost setting is δ < c < N−2
2 δ
range where the star is efficient, but not stable. The value of linking to other players j, wij, is set to 1000; the cost of a link is 1000; δ was set to 0.8. Pairwise stable is any minimal connected network. Three different groups played the game cooperatively by discussing possible solutions and agreeing on the links they form. A first treatment allows for side-payments to compensate those players bearing larger costs; the second is without side payments. With side-payments two groups coordinate on efficient outcomes, while the third group forms a ring. In the second treatment, there are no side-payments. The first two groups coordinated on the line network. The other group, however, did not consider to agree on an unequal outcome and coordinated on a ring, splitting the cost equally. This result is remarkable insofar as the line is the pareto-optimal outcome: While the ring provides an equal payoff of 240 to all players, the line provides a payoff of 240 to the players with 2 links, but the two extreme players get 952. This agreement was reached tossing a coin. Such an outcome requires that players accept inequality that the players distinguish between the opportunity to gain more before the game starts, and the actual outcome.
Falk and Kosfeld (2003) consider the BG game with 1-way and 2-way flows of benefit and no decay. There are four players in the game. The cost settings cover empty, minimal and star networks as the equilibrium prediction. The game is played for five rounds. Links are formed simulta- neously. After a step, players are informed about the network, costs and the connected players. They find that
– In 1-way flow models many outcomes are Nash strict Nash equilibria (between 40 and 60 %). However, for the 2-way flow model, there is no strict Nash equilibrium, and fewer Nash equilibria (between 10 and 30 %).
– If there is more than one unique stable network, subjects solve the coordination problem by opting for the efficient network.
– Higher costs support the selection of both Nash and strict Nash so- lutions in the 1-way flow model, but have a negative impact on the selection in the 2-way flow model.
Falk and Kosfeld (2003) provide two possible explanations for the unequal results in the 1-way and 2-way flow models. The first possibility is the asym- metry in payoffs: In the 1-way flow model, the ring is the stable network, as each player has to create a link to participate in the value of the network. Costs and benefits are distributed equally. In the 2-way flow model, the sta- ble solution is the centre-sponsored star, but no rational player wants to be in this position. Their data support this hypothesis, as they find that when such solutions are reached, they are unstable, i.e. the disadvantaged players sever their links. The other possible explanation offered are social prefer- ences. This hypothesis is supported by their finding that the frequency of Nash outcomes decreases the more unequal the payoffs are - this becomes especially apparent in the low frequency of the centre-sponsored star. Using a regression model, they find that individuals are more likely to revise their strategy if outcomes were unequal.
Using a similar setup asFalk and Kosfeld(2003),Bernasconi and Galizzi (2005) find very different results. They consider four treatments with low and high costs and one- and bi-directional flow of benefits. The main differ- ence to the former experiment is a more neutral labelling. Bernasconi and Galizzi(2005) claim that the use of ordered labels A,B,C,D in Falk and Kos- feld’s experiment serves as a coordinating device, as they find in their own experiments that the ring from A to D can be observed significantly more often than when random labels are used. They therefore choose instead
more neutral labels like ’&’ or ’%’. They find that in the one-directional treatments almost no Nash networks emerge (between 1% and 3%). In the bi-directional experiments sometimes Nash networks form, but also with comparatively low frequency (between 13% and 17%).
Callander and Plott (2005) consider a BG model with 1-way flow of benefits with no decay. They consider different treatments with homogenous and heterogeneous cost settings. Cost settings are such that the wheel is strict Nash. For the homogenous case, they find that
– The empty network never occurs.
– If networks converge, it is usually a Nash equilibrium, however, not strict.
– Not all Nash equilibria are stable, often an equilibrium state collapsed again.
Looking at how players take decisions and the dynamics of behaviour, they find that
– Players do typically not play myopic best-responses as in the BG model. They often use simple strategies considering the future out- comes of the game. Agents make more sophisticated decisions antici- pating future outcomes.
– Agents using such simple strategic behaviour follow their strategy more consistently.
– Convergence depends on the behaviour of all agents. The more agents switch to simple strategic behaviour, the more likely the network con- verges.
– The more agents remain committed to their behaviour, the more likely other agents will adopt this behaviour as well.
Conte et al (2009) investigate a link formation game where links are formed only if both players agree. In each round of the game players bid for links simultaneously. The main interest is not whether networks con- verge, but which individual strategies are responsible for the result. There are six players, no decay, and cost settings are such that the equilibrium prediction is a minimal connected network. Subjects have full information about the network. In total, there were 54 participants. Nine experimental sessions were run with six players per session. A session lasts at least 15 rounds, after which a random generator determined to stop the session. In the experiments, minimally connected networks emerge; however, stability is low. Conte et al (2009) attribute this to the fact that many equilibria are possible, so that it is difficult to coordinate on a certain outcome. They also observe that when a minimal connected network is established, some players are tempted to experiment with alternative strategies. As a result, a network might come out of equilibrium again. From the individual per- spective, they find that 40 % of strategies are best-response strategies. The remaining 60% strategies are not very far from best-response behaviour. Distance is determined by calculating an index based the difference be- tween profits of actual and best response behaviour. Common alternative strategies are reciprocator and opportunistic behaviour. The first behaviour maximises direct connections by always offering links to those players who offered links in the previous round. The second behaviour tries to maximise indirect links by removing direct links whenever possible. Best response behaviour is strongly group driven, i.e. the more players adopt this strat- egy the more likely that the remaining agents follow. There is an overlap between best response and the other strategies. Conte et al(2009) estimate
econometrically that 42% of players belong to the opportunistic, 31% to the best response type and 27% to the reciprocator type. The high portion of the opportunistic type thus points, similar as the previous studies, to more complex than myopic best response behaviour.
Goeree et al (2009) test whether heterogeneous players manage better to agree on efficient networks. They consider three treatments: A base- line treatment with homogenous agents, a treatment with a low-cost agent (experiencing lower costs for maintaining a link), and a treatment with a high-value agent (experiencing and providing higher utility per direct or in- direct link). They find that with homogeneous agents, formation of equilib- rium networks fails. Introducing cost heterogeneity supports the emergence of equilibrium networks in the form of minimal connected or star-networks. When agents receive different value from linking the chance to observe equi- librium networks is highest.
Summarising the main results of the experimental literature, the follow- ing conclusions can be drawn:
– The frequency of equilibrium networks differs strongly between the experiments. Some authors find no Nash networks at all. Maximum rates observed go up to 40%.
– Even where Nash networks are found as good predictors, it becomes apparent that the actual individual decisions deviate from the myopic best response (Callander and Plott 2005). Basic strategies like oppor- tunistic linking, reciprocating behaviour or simple strategic-decision making are more common.
– The more agents commit to a certain behaviour, the more likely con- vergence.
– Some authors further mention an equality norm, i.e. a preference of the players for equal distributions of cost (e.g. Vanin 2002).