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Private Blotto: Viewpoint Competition with Polarized Agents

Kate Donahue, Jon Kleinberg

TL;DR

Private Blotto game is proposed and studied, a variant with the key difference that individual agents act independently, without being coordinated by a central "Colonel", and it is completely characterize the Nash stability of this game and how this impacts the amount of "misallocated effort" of users on unimportant items.

Abstract

Social media platforms are responsible for collecting and disseminating vast quantities of content. Recently, however, they have also begun enlisting users in helping annotate this content - for example, to provide context or label disinformation. However, users may act strategically, sometimes reflecting biases (e.g. political) about the "right" label. How can social media platforms design their systems to use human time most efficiently? Historically, competition over multiple items has been explored in the Colonel Blotto game setting (Borel, 1921). However, they were originally designed to model two centrally-controlled armies competing over zero-sum "items", a specific scenario with limited modern-day application. In this work, we propose and study the Private Blotto game, a variant with the key difference that individual agents act independently, without being coordinated by a central "Colonel". We completely characterize the Nash stability of this game and how this impacts the amount of "misallocated effort" of users on unimportant items. We show that the outcome function (aggregating multiple labels on a single item) has a critical impact, and specifically contrast a majority rule outcome (the median) as compared to a smoother outcome function (mean). In general, for median outcomes we show that instances without stable arrangements only occur for relatively few numbers of agents, but stable arrangements may have very high levels of misallocated effort. For mean outcome functions, we show that unstable arrangements can occur even for arbitrarily large numbers of agents, but when stable arrangements exist, they always have low misallocated effort. We conclude by discussing implications our results have for motivating examples in social media platforms and political competition.

Private Blotto: Viewpoint Competition with Polarized Agents

TL;DR

Private Blotto game is proposed and studied, a variant with the key difference that individual agents act independently, without being coordinated by a central "Colonel", and it is completely characterize the Nash stability of this game and how this impacts the amount of "misallocated effort" of users on unimportant items.

Abstract

Social media platforms are responsible for collecting and disseminating vast quantities of content. Recently, however, they have also begun enlisting users in helping annotate this content - for example, to provide context or label disinformation. However, users may act strategically, sometimes reflecting biases (e.g. political) about the "right" label. How can social media platforms design their systems to use human time most efficiently? Historically, competition over multiple items has been explored in the Colonel Blotto game setting (Borel, 1921). However, they were originally designed to model two centrally-controlled armies competing over zero-sum "items", a specific scenario with limited modern-day application. In this work, we propose and study the Private Blotto game, a variant with the key difference that individual agents act independently, without being coordinated by a central "Colonel". We completely characterize the Nash stability of this game and how this impacts the amount of "misallocated effort" of users on unimportant items. We show that the outcome function (aggregating multiple labels on a single item) has a critical impact, and specifically contrast a majority rule outcome (the median) as compared to a smoother outcome function (mean). In general, for median outcomes we show that instances without stable arrangements only occur for relatively few numbers of agents, but stable arrangements may have very high levels of misallocated effort. For mean outcome functions, we show that unstable arrangements can occur even for arbitrarily large numbers of agents, but when stable arrangements exist, they always have low misallocated effort. We conclude by discussing implications our results have for motivating examples in social media platforms and political competition.
Paper Structure (24 sections, 47 theorems, 201 equations, 2 figures, 2 algorithms)

This paper contains 24 sections, 47 theorems, 201 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modeling decentralized conflicts.
  3. Motivating examples and further related work
  4. Overview of results
  5. Model
  6. Outcome functions
  7. Agent cost
  8. More agents: $N \geq M$
  9. Median outcome
  10. Mean outcome
  11. Misallocated effort
  12. More items: $M>N$
  13. Discussion
  14. Further related works
  15. Colonel Blotto
  16. ...and 9 more sections

Key Result

Lemma 1

If there are more agents than items ($N \geq M$) and the cost for leaving a item empty is sufficiently high, then no item will be left empty, regardless of if median or mean outcome function is used. Specifically, this occurs when: Moreover, agent strategy becomes independent of biases $\beta_a, \beta_b$ and relies solely on the number of agents of each type on each item, $\{a_i, b_i\}, \ i \in [

Figures (2)

  • Figure 1: Figure illustrating when stable arrangements of agents onto items fail to exist for mean outcome ($M=2$ items). For clarity, only displayed for $N_a \geq N_b$.
  • Figure 2: Version of Figure \ref{['fig:stable']} illustrating when stable arrangements of agents onto items exist for mean outcome ($M=2$ items), but with differing weights on the $M=2$ items. For clarity, only displayed for $N_a \geq N_b$.

Theorems & Definitions (76)

  • Definition 1: Colonel Blotto
  • Definition 2: Private Blotto
  • Definition 3: Nash stability (pure)
  • Lemma 1
  • Definition 4: Median-critical region
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • ...and 66 more