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Strategic Negotiations in Endogenous Network Formation

Akhil Jalan, Deepayan Chakrabarti

TL;DR

An efficient algorithm for finding the set of Nash equilibria, if any exist, and certify their nonexistence otherwise is provided and an algorithm by which new agents can learn the information needed for strategic behavior is developed.

Abstract

In network formation games, agents form edges with each other to maximize their utility. Each agent's utility depends on its private beliefs and its edges in the network. Strategic agents can misrepresent their beliefs to get a better resulting network. Most prior works in this area consider honest agents or a single strategic agent. Instead, we propose a model where any subset of agents can be strategic. We provide an efficient algorithm for finding the set of Nash equilibria, if any exist, and certify their nonexistence otherwise. We also show that when several strategic agents are present, their utilities can increase or decrease compared to when they are all honest. Small changes in the inter-agent correlations can cause such shifts. In contrast, the simpler one-strategic-agent setting explored in the literature lacks such complex patterns. Finally, we develop an algorithm by which new agents can learn the information needed for strategic behavior. Our algorithm works even when the (unknown) strategic agents deviate from the Nash-optimal strategies. We verify these results on both simulated networks and a real-world dataset on international trade.

Strategic Negotiations in Endogenous Network Formation

TL;DR

An efficient algorithm for finding the set of Nash equilibria, if any exist, and certify their nonexistence otherwise is provided and an algorithm by which new agents can learn the information needed for strategic behavior is developed.

Abstract

In network formation games, agents form edges with each other to maximize their utility. Each agent's utility depends on its private beliefs and its edges in the network. Strategic agents can misrepresent their beliefs to get a better resulting network. Most prior works in this area consider honest agents or a single strategic agent. Instead, we propose a model where any subset of agents can be strategic. We provide an efficient algorithm for finding the set of Nash equilibria, if any exist, and certify their nonexistence otherwise. We also show that when several strategic agents are present, their utilities can increase or decrease compared to when they are all honest. Small changes in the inter-agent correlations can cause such shifts. In contrast, the simpler one-strategic-agent setting explored in the literature lacks such complex patterns. Finally, we develop an algorithm by which new agents can learn the information needed for strategic behavior. Our algorithm works even when the (unknown) strategic agents deviate from the Nash-optimal strategies. We verify these results on both simulated networks and a real-world dataset on international trade.
Paper Structure (23 sections, 24 theorems, 60 equations, 7 figures, 2 algorithms)

This paper contains 23 sections, 24 theorems, 60 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Strategic Negotiations
  4. Outcomes for Multiple Strategic Actors
  5. Learning from Strategic Negotiation Outcomes
  6. Experiments
  7. Learning Experiments
  8. Negotiations on International Trade Networks
  9. Conclusions and Future Work
  10. Proofs and Additional Theoretical Results
  11. Proof of Theorem \ref{['thrm:lying-bounded']}
  12. Proof of Theorem \ref{['thrm:indiv-optimal']}
  13. Generalization to Stochastic M
  14. Proof of Proposition \ref{['prop:b-recovery']}
  15. Estimating the set of strategic agents
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $M$ be such that $M \bm{e}_i=\bm{\mu}_i$. Let $\Gamma$ be a diagonal matrix with $\Gamma_{ii}=\gamma_i$. Note that $\Gamma \succ 0$ and $\Sigma \succ 0$. There exists a unique stable point $(W, P)$: Furthermore, agents can efficiently find the stable point through honest pairwise negotiations.

Figures (7)

  • Figure 1: Network with three agents: (a) An investor trades with two hedge funds, with the investor gaining $5$ per unit contract while the hedge funds gain $1$. (b) We show the utility for the investor (left) and either hedge fund (right) for various strategic behaviors. The investor has low utility when she is honest, but is better off than the hedge funds when she is strategic.
  • Figure 2: Normalized regression error for $\hat{B}$ estimation, with shaded regions denoting $[10,90]$-percentile outcomes across $10$ independent trials. Left: $\lvert S\rvert = 0.1\times n$. Right: $d = 2$.
  • Figure 3: Balanced accuracy (the mean of the true positive rate and true negative rate) of $\hat{S}$ estimation, in the same settings as Figure \ref{['fig:regression_3wid']}. Shaded regions denote $[10,90]$-percentile outcomes across $10$ independent trials.
  • Figure 4: Effect of strategic trading on the UK (top) and Netherlands (bottom): The UK's utility is highest when it is the only strategic agent, and lowest when all are honest (the observed network). However, the pattern changes in the last quarter, where the utility when all are strategic is worse than when all are honest. The Netherlands has the highest utility when all others are honest, unlike the UK.
  • Figure 5: Nash equilibria for two agents: The utility for either agent when both are honest (solid line) is higher than when both are strategic (dashed line). When only agent P1 is strategic, P1 gains the highest utility (dotted circles) while P2's utility is lowest (dotted squares). The network settings are shown in the bottom row, with an arrow from $i$ to $j$ corresponding to $M_{ji}$.
  • ...and 2 more figures

Theorems & Definitions (47)

  • Definition 2.1: Stable point
  • Theorem 2.2: Stable network without strategy jalan-2023
  • Definition 3.1: Our Model of Strategic Contract Negotiation $(M, \Gamma, \Sigma, S)$
  • Theorem 3.2: Concave utility given others' choices
  • Corollary 3.3: Strategy yields bounded utility
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7
  • Corollary 3.8: Nash Equilibria
  • ...and 37 more