Network formation and efficiency in linear-quadratic games: An experimental study

We study five treatments varying the parameters of the linear quadratic utility function and the group size. We consider two group sizes: N=5, N=9. For each, we consider two different values of linking costs, such that for low linking costs the only Nash equilibrium network is the complete graph, while for the high linking costs there are three equilibria (empty, star and complete networks). Note that to achieve this we need to adjust the parameters of the utility function between N=5 and N=9. Therefore, we add an additional treatment to this 2x2 design for which N=9 and the payoff parameters are exactly the same as in the low cost environment of N=5, to make a clean comparison between different group sizes.

2022-08-01

2022-09-30

Effort choices of individuals
Network structure formed with a group
Payoffs of individuals

Network connectivity relative to complete network
Average network degree
Efficiency relative to equilibrium and efficient allocation

In the experiment, individuals interact in a group of N persons for t=30 rounds, groups remain unchanged (partner matching). In each round, they choose an effort level and their network neighbors by initiating links to some or all of the other N-1 group members. Links are formed unilaterally and are costly. A higher effort is more costly, individuals benefit from their own and their network neighbors' effort levels. The linear-quadratic utility function, studied in Hiller (2017) and Ballester et al. (2006), among others, describe the number of points a participant earns as a function of her choice of effort, choice of neighbors and the effort choices of their neighbors. This function is given by:

Pi_i=a*x_i-b/2*x_i^2+lambda*x_i \sum(x_j)-k*n_i

where a, b, lambda and k are parameters, x_i is the effort choice of i, n_i is the number of links initiated by i and the summation is over the neighbors of i.

Participants earn points each period in this way and their final earnings from the experiment will be the sum of points over all 30 periods. At the end of each round, they receive feedback about their own choices and payoffs, the network formed in the group and the effort choices of all group members.

We study the network structure formed, the effort choices and the payoffs of individuals and we compare them to the equilibrium values and the efficient allocation, derived in Hiller (2017) and computed numerically.

We introduce five treatments, varying group size and linking costs:
T1: N=5, k=1, lambda=0.4
T2: N=5, k=3.9, lambda=0.4
T3: N=9, k=1, lambda=0.25
T4: N=9, k=2.5, lambda=0.25
T5: N=9, k=1, lambda=0.4
Other parameters are set as a=10, b=4.

At the end of the experiment, participants fill out a survey about demographics, risk-preferences, 10-question version of big five personality test, cognitive reflection test and a dictator game measure of social preferences.

The experiment is conducted online with participants recruited on Prolific, using o-Tree as software.

References
Hiller, T. (2017). Peer effects in endogenous networks. Games and Economic Behavior, 105, 349-367.
Ballester, C., Calvó‐Armengol, A., & Zenou, Y. (2006). Who's who in networks. Wanted: The key player. Econometrica, 74(5), 1403-1417.

By office computer and the random arrival of participants to participate in the experiment on Prolific.

Group level

No

Number of groups per treatments:
T1: 15, T2: 15, T3: 10, T4: 10, T5: 10

Number of groups per treatments: T1: 15, T2: 15, T3: 10, T4: 10, T5: 10, that is 75+75+90+90+90=420 participants in total.

T1: 15, T2: 15, T3: 10, T4: 10, T5: 10,