Online Supplement for Bounded Rationality in Service Systems

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1 Online Supplement for Bounded ationality in Service Systems Tingliang Huang Department of Management Science and Innovation, University ollege London, London W1E 6BT, United Kingdom, Gad Allon Kellogg School of Management, Northwestern University, Evanston, IL 60208, Achal Bassamboo Kellogg School of Management, Northwestern University, Evanston, IL 60208, 1. Proofs of Propositions Proof of Proposition 1. Let λ n λϕ n, µ n µ, then we can treat the number of customers in the queueing system as a birth-death process with birth rate λ n and death rate µ n. We have the balance equations: λ 0 P 0 = µ 1 P 1, (λ n + µ)p n = µp n+1 + λ n 1 P n 1, n 1. Solving these equations, λ 0 λ 1...λ n 1 1 we have the limiting probabilities: P 0 = 1+Σ λ 0 λ 1...λ k 1, P n = k=1 µ k µ n (1+Σ λ 0 λ 1...λ k 1, n 1. k=1 µ k ) The necessary and sufficient condition for the existence of limiting probabilities is: Σ λ 0 λ 1...λ k 1 k=1 <. Let a µ k k = λ 0λ 1...λ k 1, using the ratio test, we have a k+1 = λ µ k k = ρϕ a k µ k 0, k. The series converges, hence, the condition is always satisfied for (0, ). Proof of Proposition 2. Define g(ϕ(p, )) e 1+e p p µ ϕ(p, )λ µ ϕ(p, )λ ϕ(p, ), then g(0) > 0 and g(d) = d < 0, where d min{1, 1 ρ }. g(ϕ(p, )) is continuous in ϕ(p, ). By the intermediate value theorem, there exists at least one ϕ (p, ) (0, d) such that g(ϕ (p, )) = 0. Furthermore, g(ϕ(p, )) is strictly decreasing in ϕ(p, ). Therefore, the solution is unique. Proof of Proposition 3. (i) To prove this part, we need Lemma E.1 in Appendix B of Huang et al. (2012), which states that the equilibrium joining fraction is monotone decreasing in the level of bounded rationality if the joining fraction is above one half. The reason we need Lemma E.1 is the following: The fact that ϕ(p, ) > 1 2 to p µ 1 2 is equivalent to p λ > 0, i.e., p < p. Parts (ii) and (iii) can be shown similarly. ec1 µ ϕ(p, > 0, which is equivalent )λ

2 ec2 (iv) For fixed, denote F (p, ϕ(p)) p µ ϕ(p)λ e p µ ϕ(p)λ 1+e to F (p, ϕ(p)) = 0. For convenience, we denote f f(p, ϕ(p)) = p ϕ(p), then equation (2) is equivalent µ ϕ(p)λ function theorem, we take first derivative w.r.t. p in equation F (p, ϕ(p)) = 0. We have Simplifying the equation above, we obtain This completes the proof. e f (1 + e f ) + e f λϕ (p) 2 (1 + e f ) 2 (µ ϕ(p)λ) + 2 ϕ (p) = 0. ϕ e f (µ ϕ(p)λ) 2 (p) = (1 + e f ) 2 (µ ϕ(p)λ) 2 + e f λ < 0.. Using the implicit Proof of Proposition 4. We first show that p () = p (0) ɛ for some ɛ > 0 when is strictly positive but sufficiently small. We exhaust all candidates to prove this result. First, we show that p () cannot be equal to p (0) when is strictly positive but sufficiently small. By Lemma E.13 in Huang et al. (2012), Π(p (0), ) < Π nr, and Π(p (0), 0) = lim 0 Π(p (0), ) < Π nr, for (0, ) for some > 0. Hence, we have Π(p (0), ) in the neighborhood of Π(p (0), 0) when is small by continuity. Now if we charge price p = p ɛ = nr µ ɛ for some small ɛ > 0, then under full rationality, the customer who sees n r 1 customers in front of him will join the queue with ɛ utility. With the level of bounded rationality, his joining probability would be ϕ nr 1 = < 1 but can be sufficiently close to 1, which implies less congestion and thus lower revenue. We have Π(p ɛ, ) < Π(p ɛ, 0) < Π nr, for (0, ɛ ) for some ɛ > > 0. However, we know that lim 0 Π(p ɛ, ) = Π(p ɛ, 0), and lim ɛ 0 Π(p ɛ, 0) = Π nr. Hence, Π(p ɛ, ) can be made arbitrarily close to Π nr when is small and ɛ is also small. Hence, we have Π(p (0), ) < Π(p ɛ, ), when is small and ɛ is also small. This shows that p (0) cannot be the optimal price when customers are slightly boundedly rational. Next we show that any price taking this form p = p + ɛ = nr µ ɛ e ɛ 1+e + ɛ for some fixed small ɛ > 0 cannot be the optimal price either. Under full rationality, the customer who observes n r 1 customers in front of him will not join the queue, and the revenue is strictly higher if the price p 1 = (nr+1) µ is charged instead since this modification will still induce the same number of customers to join and the revenue per customer is strictly higher. We have Π(p + ɛ, 0) < Π nr+1 Π nr. Furthermore, we have Π(p + ɛ, ) < Π nr+1 for (0, ɛ1 ) for some ɛ1 > 0. Hence, Π(p + ɛ, ) < Π(p ɛ 1, ) when is small and ɛ 1 is also small. Other prices faraway from p (0) clearly cannot be the optimal price when is small. Therefore, p () = p (0) ɛ for some ɛ > 0 when is small.

3 Finally, we need to verify the existence of the optimal price p (). For any > 0, we know Π(p, ) is continuous over the closed interval [p (0) µ, p (0)]. Hence, there exists some p () [p (0) µ, p (0)] to maximize Π(p, ). We have shown that the optimal price under a little bit of bounded rationality is strictly lower than the optimal price under full rationality. Now we can prove the second part of the proposition by noting the following: Π(p (), ) = Π(p ɛ, ) < Π(p ɛ, 0) < Π nr, for (0, ɛ ) for some ɛ > 0. Proof of Proposition 5. (i) ecall that Π I (p, ) = pϕ(p, )λ, where ϕ(p, ) is the unique solution to equation (2). For any fixed, we shall abuse notation by writing Π I (p) and ϕ(p) for brevity. Taking first derivative, we have Π I (p) = λ[ϕ(p) + pϕ (p)], where denotes the derivative. We have shown that ϕ (p) < 0. Let h(p) = 0, we have p = ϕ(p) > 0. Now we investigate whether ϕ (p) this necessary FO has multiple solutions. Substituting ϕ (p), we have p = λϕ(p) (µ ϕ(p)λ) ϕ(p). We are interested in whether this equation has a unique solution. For exposition convenience, we denote g(p) λϕ(p) (µ ϕ(p)λ) 2 + ec3 p, then the question is whether g(p) = 0 has a unique solution. 1 ϕ(p) We claim that g(p) is strictly decreasing in p. It is clear that the first term in the HS λϕ(p) (µ ϕ(p)λ) 2 is strictly decreasing in p and so are the second and third terms. Hence g(p) is strictly decreasing in p. Note g(0) > 0, and g( ) =, so there exists a unique p such that g(p ) = 0. Finally, one can verify the second-order condition is satisfied. (ii) We know p ( ) solves the following equation p ( ) = λϕ(p ( ), ) (µ ϕ(p ( ), )λ) ϕ(p ( ), ). Using the implicit function theorem and after simplifying, we have Note that A > 0 and ϕ(p, ) p p ( ) = A ϕ(p, ) + (µ ϕ(p, )λ) 3 (1 ϕ(p, )). (µ ϕ(p, )λ) 3 (1 ϕ(p, )) 2 A ϕ(p, ) p < 0, we know the denominator is strictly positive. Hence, p ( ) has the same sign as the numerator. If ϕ(p, ) 0, i.e., p ( ) p (by Proposition 3), then p ( ) > 0. Otherwise, the numerator can be negative depending on the parameters. Hence, p ( ) > 0 if p ( ) p; Otherwise, p ( ) has the same sign as A ϕ(p, ) A λ(µ + λϕ(p, ))(1 ϕ(p, )) 2 + (µ ϕ(p, )λ) 3. + (µ ϕ(p, )λ) 3 (1 ϕ(p, )), where To further simplify the result in terms of primitives such as, we want to know whether the price p (which leads to equilibrium joining fraction 0.5 regardless of the level of bounded rationality)

4 ec4 can be optimal. Suppose it were, then we have the condition by plugging it into the condition the optimal price has to satisfy, Simplifying yields 2 2µ λ = 2λ (2µ λ) = 1 2 2µ (2µ λ) 2. If it is positive, i.e., 4µ, we know dp ( 0 ) > 0, since ϕ(p, 0) (2µ λ) 2 d = 0. Then, it is clear that for any > 0, we have dp ( ) d > 0. On the other hand, if < 4µ, so that (2µ λ) 2 0 < 0, one can easily compute p (0) = (1 ). If µ p (0) p, which is equivalent to 4µ, we have dp (0) (2µ λ) 2 This then implies that dp ( ) d > 0 for any [0, ). Proof of orollary 1. Using the envelope theorem, we have dπ I (p ( ), ) d = p ( )λ ϕ(p ( ), ). Then all the results (i), (ii) and (iii) follow directly from Proposition 3 (i)-(iii). d > 0. Proof of Proposition 6. By definition, Π I (λ) = p[ϕ (λ)λ + ϕ(λ)]. To determine its sign, we want to first determine the sign of ϕ (λ). There are at least two ways to do this. First, observe the equilibrium condition, equation (2). Suppose ϕ(λ) is increasing in λ, then the LHS is decreasing while the HS is increasing, a contradiction. Hence, ϕ (λ) < 0. The other way is to derive this derivative using the implicit function theorem. For convenience, denote f f(λ) = p and F (λ, ϕ(λ)) derivative and simplifying, we have which clearly implies and dϕ (λ) dλ < 0. µ ϕ(λ)λ, ef ϕ(λ). The equilibrium condition amounts to F (λ, ϕ(λ)) = 0. Taking first 1+e f e f ϕ (λ)λ + ϕ(λ) (1 + e f ) 2 (µ ϕ(λ)λ) + 2 ϕ (λ) = 0, dϕ (λ) dπ I (λ) < 0, dλ dλ Proof of Proposition 7. To show this proposition, we study the social welfare function W () as is strictly greater than but arbitrarily close to 0, and compare it with W (0). We start from the case when only the customers on the two marginal states on the positive and the negative side randomize. Let σ() be the probability of joining for the customer who sees n s 1 customers in the queue in front of him, and δ() be the probability of joining for the customer who observes n s customers in the queue. We omit for simplicity. Let u 0 U ns 1 and u 1 U ns be their expected utilities of joining respectively.

5 If n s > µ 1, we have u 2 0 0, u 1 0, u 0 +u 1 0. By Lemma E.10 which gives conditions under which less congestion implies more welfare holds for any number of customers joining using logit probabilities, we need to show ec5 (1 σ)ρ ns + σ(1 δ)ρ ns+1 + σδρ ns+2 < ρ ns+1, (E.1) when is small. Some algebra tells us that it suffices to show e u0 M() σ(δρ + 1) = 1 + e u 0 (ρ e u e u 1 + 1) < 1. We want to know the sign of M () when is small. After lengthy algebra, we have M ()(1 + e u 0 ) 2 (1 + e u 1 ) 2 2 e u 0 when (0, 1), where M ( 1) = 0. = ρu 1 e u0+u1 (ρ + 1)u 0 e 2u 1 [(ρ + 2)u 0 + ρu 1 ]e u 1 u 0 < 0, Observing this inequality, we can see that, if u 0 + u 1 > 0, u 0 > 0, u 1 < 0, i.e., n s < µ 1 2, then lim M () > 0. Hence, the social welfare will decrease in this case. If u 0 + u 1 = 0, u 0 > 0, u 1 < 0, 0 i.e., n s = µ 1, then lim M () > 0. If u 2 0 = 0, u 1 < 0, then M () has the same sign as 2ρu 1 e u1 0 when is close to 0, i.e., lim 0 M () > 0. Next, let us consider the case when the customers in the two marginal states on the positive side and two marginal states on the negative side join with some positive probabilities. Let σ 1 (), σ(), δ(), δ 1 () be the probabilities of joining for the customers who observe n s 2, n s 1, n s, n s + 1 customers respectively. We will also omit their dependence on for brevity hereafter. We want to show (1 σ 1 )ρ ns 1 + σ 1 (1 σ)ρ ns + σ 1 σ(1 δ)ρ ns+1 + σ 1 σδ(1 δ 1 )ρ ns+2 + σ 1 σδδ 1 ρ ns+3 < ρ ns+1, (E.2) when is small. One obvious way to show this inequality is to use the similar technique, differentiation, as for the case where two customer join with non-degenerate probabilities. But it turns out to be untractable. Here is the technique we follow: We already know when (0, 1), Inequality (E.1) holds. Then, it suffices to show (1 σ 1 ) + σ 1 ρ 2 σ 1 σδδ 1 ρ 3 + σ 1 σδδ 1 ρ 4 < ρ 2 which is equivalent to σ 1 σδδ 1 1 σ 1 < ρ2 1 ρ 3 (ρ 1).

6 ec6 This inequality can clearly be satisfied for (0, 2), where 2 makes the inequality above equal. Hence, when (0, min{ 1, 2}), inequality (E.2) is satisfied. Before we generalize our result, let us also consider the case when the customers in the three marginal states on the positive side and three marginal states on the negative side join with nondegenerate probabilities. Let σ 2 (), σ 1 (), σ(), δ(), δ 1 (), δ 2 () be the probabilities of joining for those who see n s 3, n s 2,..., n s + 2 customers respectively. We need to show (1 σ 2 )ρ ns 2 + σ 2 (1 σ 1 )ρ ns 1 + σ 2 σ 1 σ(1 σ)ρ ns + σ 2 σ 1 σ(1 δ)ρ ns+1 + σ 2 σ 1 σδ(1 δ 1 )ρ ns+2 + σ 2 σ 1 σδδ 1 (1 δ 2 )ρ ns+3 + σ 2 σ 1 σδδ 1 δ 2 ρ ns+4 < ρ ns+1, (E.3) when is small. We know, when (0, min{ 1, 2}), inequality (E.2) is satisfied. Hence, it suffices to show which is equivalent to 1 σ 2 + σ 2 ρ 3 σ 2 σ 1 σδδ 1 δ 2 ρ 5 + σ 2 σ 1 σδδ 1 δ 2 ρ 6 < ρ 3. σ 2 σ 1 σδδ 1 δ 2 1 σ 2 < ρ3 1 ρ 5 (ρ 1). This inequality can be satisfied for (0, 3), where 3 Hence, when (0, min{ 1, 2, 3}), inequality (E.3) is satisfied. makes the inequality above be equality. learly, we can proceed as this until the first arrival customer joins with a positive probability, i.e., 2n s customers join with positive probabilities. For the case when any 2n + 2 customers join with positive probabilities, n n s 1, we have the inequality to be satisfied σ n...σ 2 σ 1 σδδ 1 δ 2...δ n 1 σ n < ρn+1 1 ρ 2n+1 (ρ 1). (E.4) This inequality can be satisfied for (0, n), where n makes the inequality above be equality. Hence, when (0, min{ 1, 2, 3,..., n}), we are done. Now, we need to consider the cases, when the customers in the 2n marginal states join with some positive probabilities, where n > n s. For example, for n = n s + 1, we need to show (1 σ ns 1)ρ + σ ns 1(1 σ ns 2)ρ 2 + σ ns 1σ ns 2σ(1 σ ns 3)ρ σ ns 1σ ns 2...σδδ 1...(1 δ ns 1)ρ 2ns + σ ns 1σ ns 2...σδδ 1...δ ns 1(1 δ ns )ρ 2ns+1 + σ ns 1σ ns 2...σδδ 1...δ ns 1δ ns ρ 2ns+2 < ρ ns+1 (E.5) when is small. Let x() (σ 1, 1) be such that x() = y + (1 y)σ 1, where y (0, 1), then it is easy to verify that inequality (E.3) can be modified to (1 σ 2 )ρ ns 2 + σ 2 (1 σ 1 )ρ ns 1 + σ 2 σ 1 σ(1 σ)ρ ns + σ 2 σ 1 σ(1 δ)ρ ns+1 + σ 2 σ 1 σδ(1 δ 1 )ρ ns+2 + σ 2 σ 1 σδδ 1 (1 δ 2 )ρ ns+3 + σ 2 σ 1 σδδ 1 δ 2 ρ ns+4 < x()ρ ns+1 (E.6)

7 ec7 when (0, 1 ) for some 1 > 0. In general, inequality (E.4) can be modified to Then, it suffices to show σ n...σ 2 σ 1 σδδ 1 δ 2...δ n 1 σ n < yρn+1 1 ρ 2n+1 (ρ 1). δ ns ρ 2ns+1 (ρ 1) < (1 y)ρ ns+1, (E.7) (E.8) which is equivalent to δ ns < 1 y ρ n s (ρ 1), which can clearly be satisfied for (0, y ) for some y > 0. (E.9) When the number of marginal states in which the customers randomize goes to infinity (by Proposition 1, the system is always stable), then we need to show the following summable series is less than ρ ns+1 : (1 σ ns 1)ρ + σ ns 1(1 σ ns 2)ρ 2 + σ ns 1σ ns 2σ(1 σ ns 3)ρ σ ns 1σ ns 2...σδδ 1...(1 δ ns 1)ρ 2ns + [σ ns 1σ ns 2...σδδ 1...δ ns 1(1 δ ns )ρ 2ns+1 + σ ns 1σ ns 2...σδδ 1...δ ns 1δ ns (1 δ ns+1)ρ 2ns+2 + σ ns 1σ ns 2...σδδ 1...δ ns δ ns+1(1 δ ns+2)ρ 2ns ] < ρ ns+1. (E.10) We can show that the part in [.] can be made less than (1 y)(1 σ 1 )ρ ns+1 as (0, ε ) as follows: σ ns 1σ ns 2...σδδ 1...δ ns 1(1 δ ns )ρ 2ns+1 + σ ns 1σ ns 2...σδδ 1...δ ns 1δ ns (1 δ ns+1)ρ 2ns+2 + σ ns 1σ ns 2...σδδ 1...δ ns δ ns+1(1 δ ns+2)ρ 2ns = ρ 2ns+1 σ ns 1σ ns 2...σδδ 1...δ ns 1[(1 δ ns ) + δ ns (1 δ ns+1)ρ + δ ns δ ns+1(1 δ ns+2)ρ ] ρ 2ns+1 (σ ns 1σ ns 2...σδδ 1...δ ns 1)[1 + δ ns ρ + δ 2 n s ρ ] = ρ 2ns+1 1 (σ ns 1σ ns 2...σδδ 1...δ ns 1) 1 δ ns ρ < (1 y)(1 σ 1)ρ ns+1. (E.11) The last inequality comes from σ ns 1σ ns 2...σδδ 1...δ ns 1 (1 δ ns ρ)(1 σ 1 ) < δδ 1...δ ns 1 (1 δ ns ρ)(1 σ 1 ) < 1 y ρ ns (E.12) which can easily be satisfied as is small. The case when n s < µ 1 2 or n s = µ 1 2 can be shown by similar arguments using Lemma E.8 which states the equivalent results for the case when only the customers on the two marginal states join with logit probabilities, Lemma E.10, and Lemma E.11 in Huang et al. (2012) which give conditions under which more congestion implies less welfare holds for any number of customers joining using logit probabilities. The proofs are omitted for brevity. Hence, we have completed the proof.

8 ec8 Proof of Proposition 8. Given Lemma E.12 in Appendix B of Huang et al. (2012) which shows that for any price charged in the interval ( (n 0+1), n 0 ], the conclusion holds, we only need µ µ to show when p is outside of the interval ( (n 0+1), n 0 ], the conclusion continues to hold. µ µ We use the same argument as Lemma E.12 in Huang et al. (2012). We know that W n0 is the optimal social welfare by Yechiali (1971). However, we cannot rule out the case that W (p, ) = W n0 for some p from Yechiali (1971) s results. To rule out the case, we use Haviv and Puterman (1998), who show that the only average optimal stationary policies are of control limit type, that there are at most two and, if there are two, they occur consecutively. This implies that the only gain optimal randomized stationary policies should randomize over the two control limit states if they exist. The argument is simple: for any randomized policy to be optimal, the deterministic policies it has strictly positive probabilities should yield the same average reward. In our setting with randomization using logit probabilities, their result implies that W n0 W (p, ) when > 0 since the logit joining probabilities are in the interval (0,1). is strictly larger than any Proof of Proposition 9. (i) According to Lemma E.3 which gives conditions under which the social welfare is increasing or decreasing in the level of bounded rationality in Appendix B: if µ µ λ (i). = 1 2 and p = p, then W I ( ) is constant for 0. Simplifying the conditions yields result (ii) According to Lemma E.3: If ( µ µ λ 1 2 )(p + ) > 0, then W I ( 2 2µ λ I ) strictly increases for all 0. ombining and simplifying these states in terms of p (0) and p yields the results. (iii) According to Lemma E.3: If either µ µ > 1 and p [ 2 µ µ, ), or < 1 λ 2 µ 2µ λ λ 2 and p ( 2, ], then W I ( 2µ λ µ I ) strictly decreases for all 0. ombining these cases together yields the result in (iii). (iv) According to Lemma E.3: If either µ µ µ < 1 λ 2 µ > 1 λ 2 and p < min{ 2 2µ λ, }, or µ 2 and p > max{, }, then W I ( 2µ λ µ I ) strictly increases in [0, w ) then strictly decreases in ( w, ). Again, combining and simplifying these states in terms of p (0) and p yields the results. Proof of Proposition 10. (i) Lemma E.4 in Appendix B of Huang et al. (2012) shows that the social welfare function is unimodal in the price, which allows us to invoke the first-order condition to find the optimal price. For any fixed λ > 0 and level of bounded rationality > 0, to achieve social optimality, the equilibrium joining fraction ϕ(p, ) = µ condition equation (2), we have e 1 + e p µ p = µ µ λ I µ µ λ, plugging which into the equilibrium.

9 ec9 Solving this equation, we have the unique solution p w( ) = µ log µ µ λ µ + We can easily calculate the optimal social welfare at the optimal price p w() if it is positive (which is satisfied when the conditions stated in part (i) on and are satisfied). Otherwise, we have to let price be zero to maximize the social welfare. Part (ii) follows similarly. 2. Global Stability of the Equilibrium for Invisible Queue In this section, we relax the assumption in 2.2 that each customer knows the arrival rate and the service rate. We shall show, under certain settings, how Definition 1 can emerge from customer behavior in an adaptive manner. We will next describe a model in which customers do not know the arrival rate and the service rate, and take their actions based on their past experience. We will index time period as t T {0, 1, 2, 3,...}. In period 0, each customer has no information about the service system (e.g., in terms of the expected waiting time, arrival rate, and service rate), and joins the system with probability 0.5 (or any arbitrary probability). For each period t T, we assume that the period is sufficiently long so that the system reaches its steady-state. We also assume that, within each period, using the same strategy, each customer interacts with the firm repeatedly. Let EW t EW t = 1 (µ λϕ t ) +, where ϕ t. µ denote the actual expected waiting time in period t. Therefore, is the fraction of the customers that join the system. However, each customer is boundedly rational in the sense that he does not have the perfect capability to compute EW t. Each customer thus obtains his expected waiting time estimate ÊW t EW t + ε t, where ε t is assumed to follow the logistic distribution with parameter θ as defined in 2 of the main paper. In each period t = 1, 2,..., each customer decides whether to arrive to the system or not based on his noisy estimate of the expected waiting time ÊW t 1 in the period t 1. Following the same argument in 2, the fraction of customers that join the system is ϕ t = p EWt 1 e p EW t e I which can be interpreted as the joining probability for each customer in period t = 1, 2,... In the proposition below, we shall prove that, under certain conditions, customer behavior will converge to the unique equilibrium in Definition 1. To state this proposition, we substitute the actual expected waiting time EW t = 1 (µ λϕ t ) + ϕ t = ψ(ϕ t 1 ). We then focus on the iterative equation: and define the mapping ψ from [0, 1] to [0, 1] so that

10 ec10 ϕ t = ψ(ϕ t 1 ) = p (µ λϕ t 1 ) + p. (µ λϕ t 1 ) + e 1 + e Proposition E.1. Suppose µ > λ ψ(0). If learning converges to the unique equilibrium in Definition 1. λ, customer behavior from adaptive (µ λ) 2 Proof. If =, then the fraction of the customers that arrive to the system is 0.5 independent of customer learning. It trivially converges to the equilibrium 0.5 of Definition 1. Hereafter, we focus on the case when <. There are two cases to consider. ase 1: we suppose that in period 0, µ λϕ 0 so that the system is unstable. A customer will not arrive to the system in period 1 since the estimated expected waiting time is infinity when is finite. In period 2, each customer frequents/interacts with the firm with an extremely small frequency to learn the expected waiting time, so that each customer arrives to the system with probability ϕ 2 = ψ(0). If µ λ ψ(0) > 0, then the system is stable in period 2. In period 3, each customer arrives to the system with probability ϕ 3 = ψ(ϕ 2 ) = ψ( ψ(0)) < ψ(0) since ψ(.) is strictly decreasing by the definition of ψ(.). Hence, we have µ λϕ t > 0 for t = 2, 3, 4,..., which suggests that we can focus on the mapping ψ: p µ λϕ ϕ t = ψ(ϕ t 1 ) = e t e p µ λϕ t 1 for t = 2, 3, 4,... In lemma E.1 below, we prove that ψ is contraction mapping for. Hence, let t, then the unique equilibrium lim t ϕ t = ϕ in Definition 1 will emerge. ase 2: we suppose in period t 0 = 1, 2,..., µ λϕ t0 so that the system is unstable, then one can re-label the time period so that t 0 = 0 and apply the same argument above. Lemma E.1. The function is a contraction mapping for. Proof. Denote u 1 (ϕ t ) = p p µ λϕ ψ(ϕ t 1 ) = e t e p µ λϕ t 1 µ λϕ t. Then it is easy to show that u 1(ϕ t ) = since ϕ t < 1. Note that ψ is continuously differentiable and ψ (ϕ t ) = eu 1(ϕ t )/ u 1(ϕ t ) (1 + e < λ u 1(ϕ t )/ ) 2 (µ λ). 2 λ (µ λϕ t < ) 2 λ (µ λ) 2

11 ec11 Figure E.1 onvergence to the Equilibrium ( = 1, µ = 1, λ = 0.5, = 2, p = 0.6, = 0.3) ϕ 0 = 0.5 ϕ 0 = 0.7 equilibrium ϕ = joining probability ϕt time period t Hence, for = λ, we have (µ λ) 2 ψ (ϕ t ) < 1. Therefore, there exists θ [0, 1) such that, for any x, y [0, 1], x < y, we have ψ(x) ψ(y) = ψ (ξ)(x y) θ x y, where ξ [x, y]. Hence, ψ is a contraction mapping for. To demonstrate the convergence, we provide a numerical example in Figure E.1. We use the following parameters: = 1, µ = 1, λ = 0.5, = 2, p = 0.6, = 0.3. These parameters are also used in Figure 1, 3.1. In this figure, the horizontal line denotes the equilibrium in Definition 1 which is 0.53 in this case. The dots illustrate the convergence path as a function of the time period t from the starting joining probability ϕ 0 = 0.5, while the circles for a different starting point ϕ 0 = 0.7. We can see that the customer behavior quickly converges to the equilibrium for = 0.3. Note that = 2 for this numerical example. One can give simple sufficient conditions for µ > λ ψ(0). For example, if µ λ, then regardless of price p, the condition µ > λ ψ(0) holds. 3. Additional Explanation for Proposition 7 According to equation (1), as 0, the customers who have strictly positive expected utility of joining will join the queue with probability converging to 1, and those who have strictly negative

12 ec12 expected utility will join the queue with probability converging to 0. For the sake of a thought experiment, we assume that there is a single state in which customers join with non-degenerate probabilities. If this were the case, it must be the marginal state either on the positive side, i.e., the state where the customer who observes n s 1 customers in the system, or on the negative side, i.e., the state where the customer who observes n s customers in the system. However, in the true system with boundedly rational customers, as long as > 0, there are multiple states in which customers join the system with non-degenerate probabilities. onsidering that the level of bounded rationality is close to zero, it is intuitively clear that it is the joint effect of the customer behavior in the marginal state on the positive side and the customer behavior on the negative side that determines the direction of the social welfare change. The effect of the customer behavior in the marginal state on the positive side improves the social welfare, while the effect of the customer behavior in the marginal state on the negative side is detrimental to the social welfare. We have to characterize which effect dominates the other, i.e., to disentangle the joint effect. The scenario when n s = n 0, i.e., self-interested customers bring the system to the social optimality, is rare. However, in this setting, a strictly positive and small causes each customer to join the system with non-degenerate probabilities that can only decrease the social welfare. In contrast, when n s n 0, it is the relative location of n s and µ 1 2 n s > µ that determines the result. If 1, then it is the effect of the customer behavior on the positive side of the marginal state 2 that dominates. Hence, the social welfare is improved. If n s < µ decrease the social welfare. The case when n s = µ 1 2 1, the opposite effect would 2 is more delicate since both effects come into play simultaneously. It turns out that when ρ > 1, the congestion level is so high that the negative effect dominates, and we obtain strictly lower social welfare. When ρ 1, the congestion level is low enough to allow the positive effect to dominate, and we obtain strictly higher social welfare.