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Similarity of Information in Games

Deepal Basak, Joyee Deb, Aditya Kuvalekar

Abstract

Algorithmic content targeting homogenizes information, with implications for strategic interactions. For example, this increased homogenization was arguably responsible for the run on the Silicon Valley Bank. We argue that existing measures of similarity are inappropriate for studying games -- especially coordination games -- because they do not discipline agents' conditional beliefs. We propose a class of stochastic orders, Concentration Along the Diagonal (CAD), built on agents' conditional beliefs. In canonical binary-action coordination games, greater CAD-similarity is both necessary and sufficient for strategic similarity -- agents adopt the same strategy. We further demonstrate CAD's applicability in congestion games, collective action, and second-price auctions.

Similarity of Information in Games

Abstract

Algorithmic content targeting homogenizes information, with implications for strategic interactions. For example, this increased homogenization was arguably responsible for the run on the Silicon Valley Bank. We argue that existing measures of similarity are inappropriate for studying games -- especially coordination games -- because they do not discipline agents' conditional beliefs. We propose a class of stochastic orders, Concentration Along the Diagonal (CAD), built on agents' conditional beliefs. In canonical binary-action coordination games, greater CAD-similarity is both necessary and sufficient for strategic similarity -- agents adopt the same strategy. We further demonstrate CAD's applicability in congestion games, collective action, and second-price auctions.

Paper Structure

This paper contains 32 sections, 13 theorems, 130 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Notions of Information Similarity
  4. Concentration along a Diagonal
  5. Concentration along a Diagonal: Contour Sets
  6. Comparing the notions
  7. CAD Similarity and Coordination Equivalence
  8. Binary-action private-value coordination games
  9. CAD Similarity and Coordination Equivalence
  10. Common-value Affine Coordination Games
  11. Contour-CAD and Cutoff Equilibria
  12. Applications
  13. Does more similar information improve welfare?
  14. Beyond affine coordination games
  15. Similarity and Congestion
  16. ...and 17 more sections

Key Result

Lemma 1

Let $\mathbf X$ and $\mathbf Y$ be two $\mathcal{S}^ N$-valued, exchangeable random variables with distributions $\mathcal{F}$ and $\mathcal{G}$ respectively. Moreover, $\mathcal{F}$ and $\mathcal{G}$ have identical marginals. Then, the following are equivalent.

Figures (7)

  • Figure 1: Increasing interdependence according to existing orders ($\alpha>0$).
  • Figure 2: $\mathbf Y \succcurlyeq_{cCAD} \mathbf X$ but $\mathbf Y \not \succeq_{CAD} \mathbf X$.
  • Figure 4: Correlation and Equilibrium
  • Figure : Supporting hyperplane through $y_1$ separating $\{y_1,y_2\}$ from $\{y_3\}$.
  • Figure : Supporting hyperplane through $y_1$ separating $\{y_1,y_2\}$ from $\{y_3\}$.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Definition 1: Concentration along a Diagonal
  • Lemma 1
  • Definition 2: Concentration along a Diagonal: Contour Sets
  • Definition 3: Concentration along a Diagonal: Intervals
  • Proposition 1
  • proof : Proof of Proposition \ref{['Proposition: relation between orders']}
  • Definition 4: Affine Private-value Coordination games
  • Theorem 1
  • Definition 5: Common-value Affine Coordination games
  • Definition 6
  • ...and 26 more