Online Technical Appendix Joining Longer Queues: Information Externalities in Queue Choice
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Joining Longer Queues: Information Externalities in Queue Choice Senthil Veeraraghavan • Laurens Debo Wharton School, University of Pennsylvania, 3730 Walnut St, Philadelphia, PA 19104, USA Graduate School of Business, University of Chicago, 5807 S. Woodlawn Avenue Chicago, IL 60637 [email protected] • [email protected] PROOF of Proposition 3: We show the detailed proof that Ab is not an equilibrium when fixed queue difference thresholds are greater than one, i.e. b > 1. Consider a strategy whose fixed threshold is b > 1. We write out the steady state probability transition equations for all boundary states under the Ab strategy. πk,0(λ+ μ) = πk−1,0λ+ πk+1,0μ+ πk,1μ ∀k ≥ b+ 2. πk,0(λ+ μ) = πk−1,0λg + πk+1,0μ+ πk,1μ ∀1 ≤ k ≤ b+ 1. Adding the above equations for all values of k, and writing a similar equation for all states (0, n) for all n, we have the following equations below: ρ(1− g) b ∑
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