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

Last registered on September 02, 2022


Trial Information

General Information

Network formation and efficiency in linear-quadratic games: An experimental study
Initial registration date
August 31, 2022

Initial registration date is when the trial was registered.

It corresponds to when the registration was submitted to the Registry to be reviewed for publication.

First published
September 02, 2022, 4:06 PM EDT

First published corresponds to when the trial was first made public on the Registry after being reviewed.


Primary Investigator

Duke Kunshan University

Other Primary Investigator(s)

Additional Trial Information

On going
Start date
End date
Secondary IDs
Prior work
This trial does not extend or rely on any prior RCTs.
This experiment studies network formation in a lab setting. Participants interact in a group, and choose an effort level and their connections to other group members. Effort and linking are costly but also bring benefits according to the linear-quadratic utility function studied in many network game applications. We study equilibrium formation and selection, and the efficiency of the experimental outcomes relative to the theoretical predictions. We vary the group size and the cost of linking across experimental treatments.
External Link(s)

Registration Citation

Horvath, Gergely. 2022. "Network formation and efficiency in linear-quadratic games: An experimental study." AEA RCT Registry. September 02.
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Experimental Details


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.
Intervention Start Date
Intervention End Date

Primary Outcomes

Primary Outcomes (end points)
Effort choices of individuals
Network structure formed with a group
Payoffs of individuals
Primary Outcomes (explanation)

Secondary Outcomes

Secondary Outcomes (end points)
Network connectivity relative to complete network
Average network degree
Efficiency relative to equilibrium and efficient allocation
Secondary Outcomes (explanation)

Experimental Design

Experimental Design
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.

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.
Experimental Design Details
Randomization Method
By office computer and the random arrival of participants to participate in the experiment on Prolific.
Randomization Unit
Group level
Was the treatment clustered?

Experiment Characteristics

Sample size: planned number of clusters
Number of groups per treatments:
T1: 15, T2: 15, T3: 10, T4: 10, T5: 10
Sample size: planned number of observations
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.
Sample size (or number of clusters) by treatment arms
T1: 15, T2: 15, T3: 10, T4: 10, T5: 10,
Minimum detectable effect size for main outcomes (accounting for sample design and clustering)

Institutional Review Boards (IRBs)

IRB Name
Duke Kunshan University Institutional Review Board
IRB Approval Date
IRB Approval Number


Post Trial Information

Study Withdrawal

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Is the intervention completed?
Data Collection Complete
Data Publication

Data Publication

Is public data available?

Program Files

Program Files
Reports, Papers & Other Materials

Relevant Paper(s)

Reports & Other Materials