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Private Blotto: Viewpoint Competition with Polarized Agents
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Introduces Private Blotto, a decentralized variant of Colonel Blotto where independent polarized agents choose items under median or mean outcome functions, and characterizes when pure Nash stable arrangements exist and how much misallocated effort they entail.
Problem
Modeling decentralized, self-interested agents (e.g., social media users labeling posts, political activists) who compete over multiple items with polarized viewpoints, where classical Colonel Blotto assumes a centrally coordinated colonel.
Approach
Replaces the centrally controlled Colonel Blotto setting with independent agents of two bias types who each pick one item; the outcome function (median vs. mean) determines existence of pure Nash stable arrangements and the level of misallocated effort, with a characterization of the median-critical region.
Key results
- For median outcomes, stable arrangements exist for all agent counts outside a bounded median-critical region (empty for two items; two points for three items), but worst-case misallocated effort is at least 0.25n.
- For mean outcomes, there are arbitrarily large agent counts with no stable arrangement, yet every stable arrangement has misallocated effort at most 2m (items), and is within 1 of proportional allocation per item.
- With fewer agents than items, no stable arrangement can fail to exist for n=2, while for 4 <= n < m both median and mean outcomes admit instances with no stable arrangement; a sufficiently high unlabeled cost guarantees one-agent-per-item stability.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use mean rather than median outcome aggregation | Private Blotto with more agents than items · More agents than items | misallocated effort: $2 \cd \nitem$; median outcomes: $0.25 \cd \nplayer$ lower bound | Helped |
| use mean rather than median outcome aggregation | Private Blotto with more agents than items · More agents than items | stable arrangement existence | Mixed |
| use median rather than mean outcome aggregation | Private Blotto with more agents than items · More agents than items | stable arrangement existence | Helped |
| set the empty-item cost sufficiently high | Private Blotto with fewer agents than items · More items than agents | stable arrangement existence: $\costun \geq 0.5 \cd \abs{\bias_a - \beta_{b}}$ | Helped |
| require integer rather than fractional effort allocation | Private Blotto with mean outcome · More agents than items | stable arrangement existence | Hurt |
| use fractional effort allocation | Private Blotto with mean outcome · More agents than items | stable arrangement existence: $a_i = \nplayer_a/\nitem, b_i = \nplayer_b/\nitem$ | Helped |
| use mean rather than median outcome aggregation | Private Blotto with fewer agents than items · More items than agents; $\nplayer=3$ | stable arrangement existence | Helped |
Surprising findings
- Median outcome functions guarantee stable arrangements for large agent counts but can allow stable arrangements with very high misallocated effort, while mean outcome functions can lack stable arrangements even for arbitrarily many agents.
Limitations
- Analysis restricted to pure Nash equilibria; mixed equilibria left for future work.
- Only two polarized agent types in main results; arbitrary types and weights treated mostly as extensions or hypotheses.
- Agents can only choose one item, and the model abstracts away real platform dynamics such as user incentives beyond bias-distance costs.
Experimental setup
Metrics: existence of pure Nash stable arrangements misallocated effort
Builds on: Colonel Blotto game Tullock contest success functions General Lotto
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