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Equilibria, Efficiency, and Inequality in Network Formation for Hiring and Opportunity

Cynthia Dwork, Chris Hays, Jon Kleinberg, Manish Raghavan

Abstract

Professional networks -- the social networks among people in a given line of work -- can serve as a conduit for job prospects and other opportunities. Here we propose a model for the formation of such networks and the transfer of opportunities within them. In our theoretical model, individuals strategically connect with others to maximize the probability that they receive opportunities from them. We explore how professional networks balance connectivity, where connections facilitate opportunity transfers to those who did not get them from outside sources, and congestion, where some individuals receive too many opportunities from their connections and waste some of them. We show that strategic individuals are over-connected at equilibrium relative to a social optimum, leading to a price of anarchy for which we derive nearly tight asymptotic bounds. We also show that, at equilibrium, individuals form connections to those who provide similar benefit to them as they provide to others. Thus, our model provides a microfoundation in professional networking contexts for the fundamental sociological principle of homophily, that "similarity breeds connection," which in our setting is realized as a form of status homophily based on alignment in individual benefit. We further explore how, even if individuals are a priori equally likely to receive opportunities from outside sources, equilibria can be unequal, and we provide nearly tight bounds on how unequal they can be. Finally, we explore the ability for online platforms to intervene to improve social welfare and show that natural heuristics may result in adverse effects at equilibrium. Our simple model allows for a surprisingly rich analysis of coordination problems in professional networks and suggests many directions for further exploration.

Equilibria, Efficiency, and Inequality in Network Formation for Hiring and Opportunity

Abstract

Professional networks -- the social networks among people in a given line of work -- can serve as a conduit for job prospects and other opportunities. Here we propose a model for the formation of such networks and the transfer of opportunities within them. In our theoretical model, individuals strategically connect with others to maximize the probability that they receive opportunities from them. We explore how professional networks balance connectivity, where connections facilitate opportunity transfers to those who did not get them from outside sources, and congestion, where some individuals receive too many opportunities from their connections and waste some of them. We show that strategic individuals are over-connected at equilibrium relative to a social optimum, leading to a price of anarchy for which we derive nearly tight asymptotic bounds. We also show that, at equilibrium, individuals form connections to those who provide similar benefit to them as they provide to others. Thus, our model provides a microfoundation in professional networking contexts for the fundamental sociological principle of homophily, that "similarity breeds connection," which in our setting is realized as a form of status homophily based on alignment in individual benefit. We further explore how, even if individuals are a priori equally likely to receive opportunities from outside sources, equilibria can be unequal, and we provide nearly tight bounds on how unequal they can be. Finally, we explore the ability for online platforms to intervene to improve social welfare and show that natural heuristics may result in adverse effects at equilibrium. Our simple model allows for a surprisingly rich analysis of coordination problems in professional networks and suggests many directions for further exploration.
Paper Structure (31 sections, 47 theorems, 173 equations, 4 figures)

This paper contains 31 sections, 47 theorems, 173 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related work
  4. A model of opportunity transfer in networks
  5. Opportunity creation and transfer.
  6. Notation.
  7. The network formation game.
  8. Equilibrium concept.
  9. Price of anarchy.
  10. Equilibrium and efficiency analysis
  11. Frictionless network formation.
  12. Equilibrium networks.
  13. Efficient networks.
  14. The price of anarchy.
  15. Costly network formation.
  16. ...and 16 more sections

Key Result

Proposition 4.0

In the Professional Networking Game, if $\gamma = 0$, for all $E \subseteq [n]^2$ and $(i,j) \not\in E$,

Figures (4)

  • Figure 1: A visualization of the price of anarchy in professional networking games without (left) and with (right) connection costs. Color represents the price of anarchy, where we normalize the scale as percentage above 1 (i.e., $100(\mathrm{{PoA}} - 1)$). Left: We vary over $q$ and $p$, plotting the result in \ref{['prop:indiscpoa']}. Since $p + q \leq 1$ the price of anarchy is given over the 2-dimensional simplex. Right: We vary over $p$ and $\gamma$, setting $q = 1-p$. The color displayed is the lower bound in \ref{['prop:poaedgecosts']}.
  • Figure 2: Worst-case Gini coefficient in equilibrium networks in the Professional Networking Game with connection costs $\gamma$. Color represents the a lower bound on the Gini coefficient in \ref{['prop:inequality']} achieved by setting $\delta_1 = \delta_2 = \delta_3 = 0$.
  • Figure 3: Average utility in a worst-case regular equilibrium as a function of $\gamma$ for $q = p = 0.5$, given in \ref{['prop:mechdes']}. Each discontinuity in the graph represents a point at which the set of regular equilibria changes.
  • Figure 4: A visualization of the price of anarchy in professional networking games with edge costs. The percentage that the price of anarchy is above 1 is shown as the color. Left: We vary over $p,q$. Recall that $p + q \leq 1$, so the price of anarchy is plotted over the 2-dimensional simplex. Right: As in \ref{['fig:comparison']}, the percentage that the price of anarchy is above 1 is shown by color at given values of $p$, $\gamma$ on the x- and y-axis respectively.

Theorems & Definitions (49)

  • Proposition 4.0
  • Proposition 4.0
  • Theorem 4.1
  • Definition 4.1
  • Proposition 4.1
  • Proposition 4.1
  • Definition 4.2
  • Theorem 4.3
  • Proposition 4.3
  • Theorem 5.1
  • ...and 39 more