Queueing to learn

Ma ga ia, Chia a
A icle
Queueing o lea n
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Ma ga ia, Chia a (2025) : Queueing o lea n, Theo e ical Economics, ISSN
1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 20, Iss. 2, pp. 623-665,
h ps://doi.o g/10.3982/TE4814
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Theo e ical Economics 20 (2025), 623–665 1555-7561/20250623
Queueing o lea n
Chia a Ma ga ia
Depa men o Economics, Bos on Uni e si y
I s udy he e icien design o a queue o dynamically alloca e a sca ce esou ce
o long-li ed agen s. Agen s can be se ed mul iple imes, and hei alua ions
luc ua e o e ime wi h some pe sis ence. Each agen p i a ely lea ns whe he
his p e ailing alua ion is high o low only when se ed. An agen can decide
any ime whe he o ei he join a queue o his choice o enege. I show ha i
is e icien o elici agen s’ p i a e in o ma ion by o e ing a simple bina y menu
(i.e., wo cus ome classes): a i s -come, i s -se ed queue, o a ac low- alue
agen s, and one in andom o de , o a ac high- alue agen s. When queueing
is cos ly, o e ing a single queue may be op imal because o he adeo be ween
alloca i e e iciency and he cos o sc eening.
Keywo ds. Queues, expe imen a ion, eneging, conges ion, mechanism design.
JEL classi ica ion. C73, D47, D82.
1. In oduc ion
This pape s udies he e icien design o a queue o alloca e a esou ce low. Examples o
a ioning by wai ing a e plen i ul: he alloca ion o subsidized c edi and public housing,
assignmen o homeless shel e s, p o ision o heal h and sani a ion se ices, alloca ion
o dona ed o gans, and sha ing o p ocessing powe in a capaci y-cons ained compu -
ing sys em. I s udy he design o a queue o maximize e iciency when s a egic agen s
equi e se ice epea edly and lea n hei alua ion only when se ed.
Conside he ollowing s ylized examples o which he model is sugges i e. A mi-
c o inance ins i u ion alloca es sho - e m loans o en ep eneu s o und small-scale
p ojec s, such as s a ing a business in a de eloping coun y o inc easing c op p oduc-
ion. The p o i abili y o each en ep eneu ’s p ojec depends on ma ke condi ions and
luc ua es o e ime. Each en ep eneu is unce ain abou he p ospec s o his p ojec
and can assess i s p o i abili y only when alloca ed unds o in es ; in his sense, agen s
lea n hei alua ions when hey a e se ed. Because he loans a e small and sho - e m,
he same en ep eneu eques s loans epea edly o ope a e his business when i is p o -
i able o him o do so.
Chia a Ma ga ia: [email p o ec ed]
I am indeb ed o Johannes Hö ne and La y Samuelson o hei in aluable help and suppo h oughou
his p ojec . I would like o hank Di k Be gemann, Edua do Faingold, Mi a F ick, Ryo a Iijima, and Ba Lip-
man o hei use ul commen s and sugges ions, and Gad Allon, Ba ı¸s A a, and Achal Bassamboo o insigh -
ul discussions. I bene i ed om many con e sa ions wi h isi o s a Yale, in pa icula V. Bhaska , Lau a
Do al, Ca oline Thomas, and Takuo Sugaya, and om excellen commen s om Balazs Szen es, Nikhil Vel-
lodi, Allen I. K. Vong, and a ious semina pa icipan s. I am g a e ul o Beixi Zhou o excellen esea ch
assis ance.
©2025 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE4814
624 Chia a Ma ga ia Theo e ical Economics 20 (2025)
In ui i ely, he objec i e o he designe is wo old: maximizing alloca i e e iciency
and minimizing conges ion, ha is, alloca ing he sca ce esou ce o agen s wi h he
highes expec ed alua ion (in he mic o inance example, en ep eneu s wi h he mos
p o i able in es men oppo uni ies) and educing he queue leng h ( he a e age ime
o ob ain a loan).
On he ace o i , a single i s -come i s -se ed queue is ine icien because agen s
joining he queue impose a nega i e ex e nali y on u u e a i als by inc easing he ime
i akes o u u e agen s o be se ed, and a i ing agen s may ha e a highe expec ed
alua ion. Se ing agen s in a single se ice-in- andom-o de queue de e s agen s om
joining he queue when hei p e ailing expec ed alua ion is low, alle ia ing he ex e -
nali y p oblem. In he mic o inance example, i he expec ed ne e u n om a loan is
nega i e, an en ep eneu would pos pone hei applica ion un il p ospec s imp o e i
he e is a chance o ecei ing he loan wi hou delay.
Mo e sub ly, because agen s a e se ed epea edly and lea n abou hei alua ion
when hey a e se ed, he se ice discipline also a ec s he equilib ium leng h o he
queue, o o pu i di e en ly, he p opo ion o ime ha agen s alloca e o cos ly queue-
ing. I loans a e g an ed o en ep eneu s who a e on a e age mo e op imis ic abou
hei e u n p ospec s, a la ge ac ion o hem a e likely o esubmi an applica ion
igh away, exace ba ing u u e conges ion.
The con ibu ion o his pape is o p opose a pa simonious model o in es iga e
hese adeo s and o examine a ioning by queueing in he absence o mone a y ans-
e s while conside ing he possibili y o eneging.1The op imal queueing mechanism
is ema kably simple and in ol es well-known queueing disciplines: i is a menu o a
mos wo queues, se ice is ende ed in a i s -come, i s -se ed manne in one queue
and in andom o de in he o he .
In he model, a cons an low o a esou ce is o be alloca ed o a con inuum o
o wa d-looking agen s. Capaci y is limi ed: o e any in e al o ime, only a ixed mass
o agen s can be se ed. A each momen in ime, agen s decide whe he and when
o engage in (possibly) cos ly queueing o be se ed, and each o hem can be se ed
mul iple imes. Valua ions luc ua e independen ly ac oss agen s, and each agen aces
an expe imen a ion p oblem because he lump-sum payo collec ed a each se ice e-
eals he p e ailing alua ion, which can be high o low. Ri al y gene a es an ex e nali y
p oblem, because agen s igno e he ac ha hei ac ions a ec o e all conges ion.
The se up p esen s a ew me hodological challenges. Fi s , because o eneging, he
e ela ion p inciple does no apply. Second, e en i he unde lying alua ion is bina y,
because o lea ning, an agen ’s ype belongs o a con inuum. Thi d, he lack o ans e s
p e en s he use o s anda d me hods based on he en elope heo em o elici p i a e
in o ma ion.
I o e come hese challenges by showing ha wi hou loss o gene ali y, one can e-
s ic a en ion o queues ha p o ide he agen s incen i es no o enege. This obse a-
ion, which may be use ul in o he con ex s, elies solely on he assump ion ha agen s
1Following he ope a ions esea ch e minology, I use he e m eneging o desc ibe he ac o lea ing a
queue be o e being se ed.
Theo e ical Economics 20 (2025) Queueing o lea n 625
acqui e in o ma ion abou hei alua ion only when hey a e se ed. Fu he , I p o e
ha as long as agen s’ expec ed alua ions e ol e mono onically o e ime—a condi-
ion au oma ically sa is ied in he case o bina y unde lying alua ions—a en ion can
be es ic ed o bina y menus.
I hen show ha as a as payo s a e conce ned, each queue in he menu can be
summa ized by a pai o su icien s a is ics. This pai de e mines whe he a queueing
discipline o menu o queueing disciplines is easible, ha is, whe he he induced se -
ice a e does no exceed capaci y. The idea is ha bo h payo and se ice a e depend
only on he amoun o ime an agen expec s o queue be ween wo consecu i e se ices
and on he p obabili y o ha ing a high alua ion when se ed. A e sion o he amilia
single-c ossing p ope y o p e e ences holds. Speci ically, he isk a i ude o an agen
(i.e., his p e e ence owa ds a mo e o less isky queueing discipline) depends on his
belie abou his cu en alua ion.
Ha ing cha ac e ized he op imal menu, I hen de i e a se o compa a i e s a ics
esul s. When wai ing is cos less o he esou ce is ela i ely abundan , i is e icien o
se e agen s in wo queues. In con as , when wai ing is pa icula ly was e ul, i is op i-
mal o o e a single i s -come, i s -se ed queue, because i minimizes queue leng h.2
Rela ed li e a u e This pape is ela ed o se e al s ands o li e a u e. Fi s , i belongs
o he li e a u e on s a egic beha io in queues.3The idea ha in a i s -come, i s -
se ed (FCFS) queue a ional agen s adop subop imal beha io da es back o Nao
(1969). In ha amewo k, Hassin (1985) shows ha a las -come, i s -se ed (LCFS)
queueing discipline achie es he social op imum wi hou he need o ans e s (see also
Sca sini and Shmaya (2024)). Pla z and Øs e dal (2017) ind ha in a conce queueing
game, FCFS and LCFS achie e he minimal and maximal agg ega e equilib ium payo
among all queueing disciplines, espec i ely. In hei en i onmen , as in mine, FCFS
p o ides incen i es o join he queue ea ly, which in he end hu s all agen s in equilib-
ium. The esul s in Hassin (1985)andPla z and Øs e dal (2017), howe e , ely on he
designe ’s abili y o p e en es a ing. In con as , he designe in my model canno
de ec o punish eneging and es a ing, and LCFS canno be pa o an equilib ium.
Second, he dynamic alloca ion o objec s o agen s a i ing o e ime h ough wai -
ing lis s has been s udied in he con ex o public housing and o gan ansplan s. Bo h
Leshno (2022)andBloch and Can ala (2017) conside he p oblem o alloca ing a se-
quence o he e ogeneous i ems ha a e sequen ially o e ed o agen s on a wai ing lis .
Bo h pape s assume ha he low o agen s joining he pool is exogenous, so maximizing
wel a e is equi alen o maximizing alloca i e e iciency; consequen ly, he adeo be-
ween alloca i e e iciency and conges ion is absen in hese models. Bloch and Can ala
(2017) assume ha agen s’ alua ions e ol e o e ime independen ly ac oss pe iods,
2The esul esona es wi h he anecdo al e idence ha inspi ed Milne and Olsen (2008). The au ho s
epo ha a call cen e was o e ing di e en ia ed se ices o i s wo ypes o cus ome s ( hose wi h and
wi hou a se ice-le el ag eemen equi ing a gi en pe cen age o cus ome s o be se ed wi hin a gi en
ime) only in o -peak hou s.
3The mo i a ion and modeling choices o my pape a e close o hose o Bassamboo and Randhawa
(2015), who s udy scheduling policies in a queueing sys em wi h eneging cus ome s, abs ac ing om
s a egic conside a ions.
626 Chia a Ma ga ia Theo e ical Economics 20 (2025)
and show ha se ice in a i s -come, i s -se ed o de always ou pe o ms se ice in
andom o de . Leshno (2022) assumes ha agen s’ alua ions a e cons an o e ime and
shows ha se ice in andom o de inc eases wel a e, as compa ed o i s -come, i s -
se ed o de , by pa ially shielding agen s om andom luc ua ions in wai ing ime.
Recen ly, Che and Te cieux (2023) assume ha he designe chooses bo h he queueing
discipline and he in o ma ion a ailable o he agen s,4and show ha i is op imal o
p o ide no in o ma ion abou queue leng h and se e agen s acco ding o a i s -come,
i s -se ed ule.
Thi d, he pape con ibu es o he li e a u e on dynamic mechanism design wi h
unobse able a i al. Wi h he excep ion o a ew pape s—among o he s, in he con-
ex o dynamic mechanism design wi h ans e s, Ga e (2016)andBe gemann and
S ack (2022)—mos o he li e a u e on dynamic mechanism design assumes ha he
designe obse es agen s’ a i al. The case o unobse able a i al is na u al in he con-
ex o queues: while he designe obse es agen s joining he queue, she may be unable
o de ec when an agen has balked and ejoined he queue, p esumably disguised as a
new agen . Hence, he designe is unable o condi ion on an agen ’s pas his o y o al-
loca ion so ha nei he quo a mechanisms as in Jackson and Sonnenschein (2007)no
a “quan i ied en i lemen ” mechanism as in Guo and Hö ne (2020) is easible. As a e-
sul , he designe elici s p i a e in o ma ion by le e aging agen s’ p e e ences abou he
dis ibu ion o se ice ime, ha is, hei in e empo al p e e ences.
Las , he in e ac ion be ween agen s who engage in indi idual expe imen a ion has
been s udied by he s a egic expe imen a ion li e a u e. Howe e , mos o i has ocused
on in o ma ion ex e nali ies, which a e absen in my model. An excep ion is Thomas
(2021), who analyzes conges ion ex e nali ies. C ipps and Thomas (2019) in es iga e he
in e ac ion be ween in o ma ion ex e nali ies and conges ion ex e nali ies in a queue-
ing model. Thei pape is, howe e , only angen ially ela ed o mine. In hei model,
agen s a i e o e ime and he e is a common sou ce o unce ain y, he se ice a e o
a se e . Obse a ional lea ning a ises because queue leng h and o he agen s’ eneging
decisions e eal he agen s’ p i a e in o ma ion.
The es o he pape is o ganized as ollows. Sec ion 2in oduces he model.
Sec ion 3se s up he designe ’s p oblem and simpli ies i in h ee s eps. In Sec ion 4,I
sol e he designe ’s p oblem and cha ac e ize he op imal menu o queueing disciplines.
Sec ion 5concludes he pape .
2. Se up
Time is con inuous, indexed by ≥0, and he ho izon is in ini e. A designe (she) wishes
o alloca e a pe ishable and indi isible good o a uni mass o long-li ed agen s indexed
by i∈[0, 1]. Uni s o he good a i e a a e λ,soamassλ(  − )is o be assigned o e
any in e al o ime [ , ], < .
4In my model, in o ma ion design would no help: because o he con inuum o agen s and he assump-
ion ha he queue is unobse able, agen s do no need o in e he dis ibu ion o he esidual wai ing ime
om he amoun hey ha e been wai ing.

Theo e ical Economics 20 (2025) Queueing o lea n 627
Le Ni
deno e he o al numbe o imes agen i∈[0, 1]has been alloca ed he good
in he ime in e al [0, ]( o mally, Ni
is a coun ing p ocess); easibili y equi es ha o
all ≥0andall < ,5
1
0 
dNi
di≤λ  − .(1)
The designe aims o maximize agg ega e payo s in some equilib ium o he game, as
de ined below.
Designe ’s choice The e a e no ans e s, and he good is alloca ed ia a queueing sys-
em. In o mally, he designe commi s o a menu o queues; a all imes, each agen
chooses whe he o queue and which queue o join. Queues may di e in hei capaci y,
ha is, in he amoun o esou ce alloca ed o hem, and in hei queueing discipline.
The discipline dic a es how he good is alloca ed ac oss he agen s in he queue based
on hei indi idual wai ing ime. Common examples o queueing disciplines a e i s -
come, i s -se ed (FCSF), las -come, i s -se ed (LCFS), and se ice in andom o de
(SIRO).
A menu o queues de ines an anonymous sequen ial game be ween agen s (Jo-
ano ic and Rosen hal (1988)). I ocus on s eady-s a e equilib ia. Owing o he law o
la ge numbe s, in he s eady s a e, each agen aces a single-agen decision p oblem.
The beha io o he o he agen s is ele an only inasmuch as i a ec s he wai ing ime
be o e se ice. Hence, I o malize he choice o he designe as a choice o a collec ion
o (s eady-s a e) wai ing- ime dis ibu ions.
Mo e p ecisely, a menu is a collec ion o queues Qwi h gene ic elemen q.A any
ime , an agen can ei he be queueing o no ; hence, I desc ibe agen i’s ac ion by
qi
∈ˆ
Q:=Q∪{∅}, wi h he in e p e a ion ha qi
=∅when he agen does no queue and
qi
∈Qwhen he is wai ing a some queue.
In ligh o he discussion abo e, he designe chooses a collec ion o (s eady-s a e)
wai ing- ime dis ibu ions, {Hq}q∈Q o each q∈Q,wi hHq∈H, as de ined below.6
De ini ion 1. The se His he se o cumula i e dis ibu ion unc ions, H:R+→[0, 1],
such ha H(0)=0and dH( )<∞.
Each wai ing- ime dis ibu ion Hqis he dis ibu ion o he ime an agen wai s be-
o e being se ed, p o ided ha he does no abandon he queue be o e se ice, in s eady
s a e. The equi emen ha Hqshould no ha e a oms a 0 gua an ees ha he alloca ion
is well de ined o all s a egy p o iles o he agen s. As will become clea (see Lemma 1),
he e is no loss in es ic ing a en ion o dis ibu ions ha ing ini e mean.
5To deal wi h essen ially pai wise independen p ocesses (Ni
) ≥0, one needs o wo k wi h an en ichmen
o he usual p oduc p obabili y space (see Sun (2006)). While he explici cons uc ion is omi ed, I ely on
Sun’s law o la ge numbe s o a con inuum o andom a iables.
6Fo mula ing he designe ’s p oblem as a choice o wai ing- ime dis ibu ions ci cum en s he need o
o malize he anonymous sequen ial game and he issues a ising om ha ing a con inuum o independen
con inuous- ime Ma ko p ocesses. As will become clea , i amoun s o es a ing he designe ’s p oblem as
a s a ic mechanism design p oblem.
628 Chia a Ma ga ia Theo e ical Economics 20 (2025)
To unde s and he ela ionship be ween queueing disciplines and wai ing- ime dis-
ibu ions, no e ha i a queue ope a es in a i s -come, i s -se ed manne and agen s
do no enege, in s eady s a e, all agen s wai he same de e minis ic amoun o ime be-
o e being se ed. In o he wo ds, he wai ing- ime dis ibu ion is degene a e. Simila ly,
in a se ice-in- andom-o de queue, each agen in line, i espec i ely o he amoun o
ime he has been wai ing, has he same p obabili y o being se ed in he nex ins an .
As a esul , he wai ing ime is exponen ially dis ibu ed. Fo ease o exposi ion, I es a e
he ela ionship be ween some wai ing- ime dis ibu ions and queueing disciplines in
he ollowing de ini ions.
De ini ion 2. (i) Agen s a e se ed acco ding o a i s -come, i s -se ed discipline i
he wai ing- ime dis ibu ion His a degene a e dis ibu ion.
(ii) Agen s a e se ed acco ding o a se ice-in- andom-o de discipline i he
wai ing- ime dis ibu ion His an exponen ial dis ibu ion wi h suppo [0, ∞).
(iii) Agen s a e se ed acco ding o a se ice-in- andom-o de discipline wi h a min-
imum wai ing- ime equi emen i he wai ing- ime dis ibu ion has a cons an
haza d a e and suppo [ ,∞), o some >0.
Two ema ks a e in o de . Fi s , he designe ’s choice is no es ic ed o he classes
o wai ing- ime dis ibu ions in he de ini ion abo e; see De ini ion 2.Second,while
he se ice-in- andom-o de discipline wi h a minimum wai ing- ime equi emen is a
gene aliza ion o he se ice-in- andom-o de discipline, dis inguishing be ween hem
is con enien because he la e plays a mo e c ucial ole in he op imal menu cha ac-
e iza ion.
Agen s’ ac ions A i ing agen s do no obse e agen s al eady wai ing (i.e., queue
leng h). An agen ’s s a egy speci ies when o join and lea e a queue. Because ime is
con inuous, he o mal de ini ion equi es some ca e. In pa icula , an agen who lea es
a queue a ime , ei he because he eneges o because he is se ed a ,maywan o
es a queueing wi h no delay.
In o mally, i he agen is no queueing, a (pu e) s a egy dic a es he ime a which
he joins a queue and which one. I he agen is queueing, a (pu e) s a egy speci ies
he ime a which he agen eneges, ha is, lea es he queue i by ha ime he has no
been se ed. In his case, he s a egy p esc ibes he ac ion o be aken when eneging:
ejoining a queue o no . Finally, a s a egy also p esc ibes whe he o ejoin a queue
wi h no delay a e being se ed.
Le he ime-in-queue p ocess (wi
) ≥0desc ibe he amoun o ime elapsed since
he agen las joined he queue he is cu en ly in whene e he agen is queueing; se he
ime-in-queue p ocess equal o 0 when he agen is no queueing. An agen ’s s a egy is
an impulse con ol o he p ocesses (qi
) ≥0and (wi
) ≥0.7
I he agen does no in e ene, he p ocesses e ol e exogenously as ollows. While
an agen is queueing, he ime-in-queue g ows linea ly o e ime un il he agen is
7The o mal de ini ion o s a egies as impulse con ol policies is elega ed o he Appendix.
Theo e ical Economics 20 (2025) Queueing o lea n 629
se ed, when he ime-in-queue jumps o 0. A any se ice ime, he queue p ocess
jumps o ∅. In o he wo ds, unless he agen eneges, he agen lea es he queue as soon
as he is se ed.
A any ime , he agen can in e ene and induce (qi
,wi
) o jump o ei he (∅,0
)o
(q,0
), o someq∈Q. In ui i ely, i a queueing agen lea es a queue—ei he because
he eneges o because he is se ed—and does no ejoin he queue immedia ely, he
p ocess (qi
,wi
)jumps o (∅,0
). I he agen ei he joins a queue o jockeys be ween
queues, he p ocess (qi
,wi
)jumps o (q,0
), o someq∈Q.
Payo s When alloca ed he good, agen i ecei es a lump-sum payo equal o some
s a e θi, which can ake wo possible alues, θ0and θ1,wi hθ0<0<θ
1.Agen i’s s a e
e ol es unbeknown o him acco ding o a con inuous- ime Ma ko chain (θi
) ≥0,wi h
s a e space {θ0,θ1}, ansi ion ma ix ((−ρ0,ρ0),(ρ1,−ρ1)), and ini ial p obabili y o
s a e θ1gi en by ρ0/(ρ0+ρ1). The indi idual s a e p ocesses o any pai o agen s is
assumed o be independen .
Gi en some in eg able queue p ocess (qi
) ≥0, he ealiza ion o he s a e p ocess
(θi
) ≥0, and he indi idual alloca ion p ocess (Ni
) ≥0, he ealized payo o agen iis
gi en by he long- un a e age,
limsup
T→∞
1
TT
0
θi
dNi
−T
0
c1qi
=∅ d .
This payo has wo componen s: he sum o lump-sum payo s collec ed a each con-
sump ion expe ience and he o al cos bo ne by he agen while queueing. No e he
absence o discoun ing.
S a egies and equilib ium Gi en a menu, each agen aces a single-agen Ma ko de-
cision p oblem. I in oduce a s a e a iable o desc ibe an agen ’s in o ma ion abou his
cu en alua ion. Le pi
be he belie ha agen ia aches o his alua ion being equal
o θ1. As long as he agen is no se ed, his belie abou his alua ion e ol es acco ding
o ( he i s -o de )
dpi
=1−pi
ρ0−pi
ρ1d .
Speci ically, o all ≥0andall < , such ha no se ice occu s om  o ,
pi
 =e−(ρ0+ρ1)( − )pi
+1−e−(ρ0+ρ1)( − )ρ0
ρ0+ρ1
.(2)
Equa ion (2) makes i plain ha he belie o agen iis a con ex combina ion o his pas
belie pi
and he in a ian p obabili y o θ1,ρ0/(ρ0+ρ1). Along he his o y wi h no
se ice, he pos e io belie ha he s a e is θ1con e ges o ρ0/(ρ0+ρ1). As soon as he
agen is se ed, his belie abou his alua ion jumps o 1 o 0.
I is wi hou loss o gene ali y o assume ha agen s’ s a egies a e Ma ko in cal-
enda ime, pos e io belie , cu en queue, and ime-in-queue ( ,pi
,qi
,wi
). Abusing
no a ion, I deno e by  he se o s a iona y Ma ko s a egies, which a e hose ha do
no condi ion on calenda ime.
630 Chia a Ma ga ia Theo e ical Economics 20 (2025)
As explained abo e, he agen ’s p oblem is o malized as an impulse con ol: he
agen chooses he ( andom) da es a which he in e enes and adjus s his ac ion, ha is,
he da es a which he joins o lea es a queue, in addi ion o choosing whe he o ejoin
a queue immedia ely a e eneging o being se ed. No ice ha , gi en an ini ial s a e
(p,q,w), unless he agen adjus s his ac ion by joining o lea ing a queue, he e olu ion
o he belie pis a su icien s a is ic o he ime-in-queue e olu ion. Hence, he e is no
loss o gene ali y in ocusing on impulse-con ol policies ha a e Ma ko in he belie .
I ocus on symme ic s eady-s a e equilib ia. In a s eady s a e, each agen i∈[0, 1]
chooses his s a egy σi o maximize
Vσi,{Hq}q∈Q:=Eσi,{Hq}q∈Qlimsup
T→∞
1
TT
0
θi
dNi
−T
0
c1qi
=∅ d .(3)
De ini ion 3. A symme ic s eady-s a e equilib ium is a pai (σ,{Hq}q∈Q),σ∈such
ha he s a egy σ∈Sigma is op imal gi en {Hq}q∈Q⊂H.
The designe ’s goal is o choose a queueing menu o maximize agg ega e payo s in
some equilib ium o he game. Mo e p ecisely, she chooses a symme ic s eady-s a e
equilib ium (σ,{Hq}q∈Q) o maximize agg ega e payo s. Because, by de ini ion, he
equilib ium is symme ic, each agen achie es he same ealized payo (3). Hence, he
agg ega e payo equals he payo o a ep esen a i e agen , deno ed by ihe ea e .
The designe aces he agg ega e capaci y cons ain (1). In he spi i o he law o
la ge numbe s, I s a e he capaci y cons ain as a bound on he a e age (long- un) se -
ice a e,8
1
0Sσi,{Hq}q∈Qdi≤λ,whe eSσi,{Hq}q∈Q:=1
Ni
whe e he limi inside he in eg and is unde s ood in he sense o almos su e con e -
gence.910 Fo no a ional con enience, I d op he supe sc ip ihe ea e .
3. Designe ’sp oblem
I simpli y he designe ’s p oblem in wo s eps. Fi s , I show ha e en i an agen ’s ype
(i.e., his belie a each poin in ime) belongs o a con inuum, a en ion can be es ic ed
o bina y menus. Second, I show ha he p oblem can be cas in o a lowe -dimensional
space o su icien s a is ics.
8No e ha he law o he coun ing p ocess Ni
is join ly de e mined by {Hq}q∈Qand σi, bu o no a ional
simplici y, I keep such dependence implici .
9In Sec ion A.2.1, I show ha o any {Hq}q∈Qand any bes eply σ, he long- un se ice a e con e ges
o a cons an almos su ely.
10As I will show, he solu ion o he designe ’s p oblem in ol es wai ing- ime dis ibu ions, which a e
easily implemen able wi h well-known queueing disciplines. As a esul , i is no necessa y o a gue ha
o any equilib ium (σ,{Hq}q∈Q)∈×HQsuch ha S(σ,{Hq}q∈Q)≤λ, i is possible o ind a collec ion o
queueing disciplines implemen ing i . This is a collec ion o alloca ion ules such ha he induced anony-
mous sequen ial game be ween agen s has a symme ic equilib ium in which each playe adop s he s a -
egy σand he collec ion o wai ing- ime dis ibu ions is {Hq}q∈Q.
Theo e ical Economics 20 (2025) Queueing o lea n 637
Figu e 2. On he le panel, he se s and NBUE. On he igh panel, summa y s a is ics
o he op imal menu {H∗
0,H∗
1}and o he bes disciplines wi hin he SIRO and FCFS classes.
Solid lines depic he agen ’s indi e ence cu es; u ili y is inc easing in he sou heas di ec ion.
(θ1,θ0,c,ρ0,ρ1,λ)=(1, −3/4, 0, 1, 1, 2).
gene a ing unc ion e alua ed a −(ρ0+ρ1)and he expec ed wai ing ime. Because he
unc ion → e−(ρ0+ρ1) is con ex, he sou hwes e n bounda y o he se s co esponds o
degene a e dis ibu ions, ha is, hose assigning p obabili y 1 o some μ∈(0, ∞).The
no heas e n bounda y o NBUE co esponds o he se o exponen ial dis ibu ions,
which a e, as men ioned abo e, “ex eme” wi hin he NBUE amily. F om he pe spec-
i e o incen i es, when he wai ing ime is exponen ially dis ibu ed, he non eneging
cons ain is binding a all imes: agen s a e se ed a a cons an a e, independen o
hei a i al ime in he queue.
A no ewo hy consequence o Lemma 3is ha he classes o wai ing- ime dis ibu-
ions co esponding o he h ee classes o queueing disciplines in De ini ion 2span he
se NBUE. I agen s a e se ed acco ding o a i s -come, i s -se ed discipline, he pai
o summa y s a is ics lies on he wes e n bounda y o NBUE. I agen s a e se ed ac-
co ding o a se ice-in- andom-o de discipline, he pai o summa y s a is ics lies on
he eas e n bounda y o NBUE. Finally, each poin in he in e io o NBUE is achie ed
by a shi ed exponen ial dis ibu ion ha can be gene a ed by se ing agen s in andom
o de wi h a minimum wai ing- ime equi emen >0.
No ice, howe e , ha he cha ac e iza ion in Lemma 3doesno accoun o heca-
paci y cons ain . On he one hand, i he designe o e s a single queue and does no
disca d any o he a ailable esou ce low, he expec ed wai ing ime does no exceed
1/λ. On he o he hand, as I shall explain in he nex sec ion, iden i ying he bes ea-
sible i s -come- i s -se ed queue, o example, is no me ely a s a is ical p oblem as i
equi es analyzing he agen ’s bes eply.
4. Op imal menu
Since he designe maximizes he agg ega e payo and each agen achie es he same
payo in equilib ium, he designe ’s p e e ences coincide wi h each agen ’s p e e ences.

638 Chia a Ma ga ia Theo e ical Economics 20 (2025)
Howe e , he e is scope o he in e en ion by a designe because agen s do no in e -
nalize he ex e nali y gene a ed by hei ac ions. To shed ligh on he p oblem aced
by he designe , I now p esen a payo decomposi ion ha highligh s he sou ce o he
ex e nali y. (The o mal de i a ion can be ound in he Appendix.)
Fix a pai o wai ing- ime dis ibu ions {H0,H1}⊂Hand a s a egy σ∈NR ha
sa is ies (IC). The payo can be w i en as
Vσi,{H0,H1}=Sσi,{H0,H1}·mδH0,δH1,pσθ1−cμH1
+1−mδH0,δH1,pσθ0−cμH0,(9)
whe e m(δH0,δH1,pσ)is he long- un equency wi h which a se ice yields a lump sum
θ1,and
Sσi,{H0,H1}=1
mδH0,δH1,pσμ1+1−mδH0,δH1,pσμ0−lnpσ/(ρ0+ρ1)
is he induced se ice a e.
Acco ding o (9), hepayo om hes a egyσis he p oduc o he a e a which
he agen is se ed and he expec ed o al payo he collec s be ween se ice imes. The
la e is a unc ion o m(δH0,δH1,pσ), he p obabili y o being se ed when he s a e
is θ1. The ela ionship be ween m(δH0,δH1,pσ)and S(σi,{H0,H1})is easy o unde -
s and. F om he elemen a y enewal heo em, he expec ed se ice a e equals he in-
e se o he a e age ime be ween se ices. When joining he queue, he agen expec s
o wai an amoun o ime equal o ei he μH0o μH1, depending on he payo he e-
alized a he las se ice. Mo eo e , a e being se ed, he wai s an amoun o ime
−ln(pσ)/(ρ0+ρ1)be o e joining he queue whene e he ealized payo is θ0,which
occu s a p opo ion 1 −m(δH0,δH1,pσ)o he ime.
The cu o pσa ec s he a e a which an agen is se ed. This is a mani es a ion
o he conges ion ex e nali y, eminiscen o a “ agedy o he commons.” The designe
mus gua an ee h ough an app op ia e choice o dis ibu ions ha he se ice a e in-
duced by he agen s’ bes eply does no exceed he capaci y λ. Ideally, he designe
would like o minimize was e ul wai and pe suade he agen s wi h a low belie o delay
as long as possible be o e joining he queue.
The cu o pσalso a ec s he a e o a i al o he high ypes, o o pu i di e en ly,
he ep esen a i e agen ’s p obabili y o ealizing a high lump-sum payo when se ed.
The longe he agen wai s be o e joining he queue a e ealizing a lump sum payo
o θ0, he la ge a e o a i al o he high ypes. A he same ime, by P oposi ion 1,an
agen ejoins he queue wi h no delay as soon as he collec s a high lump-sum payo , so
a la ge a e o a i al o he high ypes may inc ease he agg ega e queueing cos .
4.1 A special case: Cos less queueing
I s a by cha ac e izing he op imal menu when queueing is cos less, o highligh he
adeo s emming om he dynamic ex e nali y p oblem. When queueing is cos less,
Theo e ical Economics 20 (2025) Queueing o lea n 639
he nega i e ex e nali y ha an agen imposes on ano he agen is no ela ed o he cos
o queuing bu a he o he i al y be ween agen s. Agen s’ desi e o be se ed can be
due o an explo a ion o exploi a ion mo i e: agen s who wan o explo e, because hey
a e g owing inc easingly op imis ic abou hei alua ion, do no in e nalize he ac ha
hey may cu ail o he agen s’ abili y o exploi , ha is, o be se ed when hei expec ed
alua ion is he highes .
To de elop some in ui ion ega ding he op imal menu, no ice ha when queueing
is cos less, se ing agen s in a single se ice-in- andom-o de queue yields a highe pay-
o compa ed o se ing hem in a single i s -come, i s -se ed queue. When agen s a e
se ed in andom o de , hey may be se ed immedia ely a e ealizing a high lump-
sum payo when hei expec ed alua ion is he highes , which ne e happens when
se ing hem in o de o a i al. The igh panel o Figu e 2plo s he summa y s a is ics
o he bes easible se ice-in- andom-o de queue and he bes easible i s -come,
i s -se ed queue: se ing agen s in andom o de may in ol e a longe wai com-
pa ed o se ing hem in o de o a i al, bu because queueing is cos less, he o me is
wel a e-imp o ing.
Now, obse e ha he designe could achie e he same payo and se ice a e as
a se ice in andom o de queue while ha ing agen s queue a all imes by o e ing a
bina y menu consis ing o a se ice-in- andom-o de queue and a se ice-in- andom-
o de queue wi h a minimum wai ing- ime equi emen . To pu i di e en ly, when
c=0, he designe does no need o y and pe suade agen s wi h a low belie o de-
lay joining he queue, and wi hou loss o gene ali y, we can assume ha in he op imal
menu, agen s queue a all imes.
Nex , I a gue ha a e sion o he amilia single-c ossing p ope y o p e e ences
holds: om he law o mo ion o belie s (7), an agen ’s a i ude owa d unce ain y in
he se ice ime—whe he he is isk-seeking o isk-a e se o e ime lo e ies—depends
on whe he he is g owing op imis ic o pessimis ic abou his alua ion. An agen wi h a
belie below he in a ian p obabili y dislikes andomness in his se ice ime, while he
opposi e is ue o agen s wi h a belie abo e he in a ian p obabili y.
Na u ally, one o he incen i e cons ain s mus bind; o o he wise, he designe
could dec ease he wai a he se ice-in- andom-o de queue and inc ease he wai
a he i s -come, i s -se ed queue and inc ease payo s, wi hou iola ing any con-
s ain . In he op imal menu, he incen i e cons ain o he agen joining wi h a low
belie binds. As a esul , he op imal menu is payo -equi alen o se ing agen s in
a single se ice-in- andom-o de queue. O cou se, o achie e he same payo wi h
a single se ice-in- andom-o de queue, he designe would need a la ge capaci y
han λ: he wai a he se ice-in- andom-o de queue in he menu is sho e han
he wai a he bes easible se ice-in- andom-o de queue (see he igh panel o
Figu e 2).
Theo em 1. Suppose c=0. Anop imalmenuisonesuch ha μH∗
1≥μH∗
0,δH∗
0≥δH∗
1,
he capaci y cons ain (C) is binding, and:
640 Chia a Ma ga ia Theo e ical Economics 20 (2025)
(i) (FCFS/SIRO menu) H∗
0is degene a e and H∗
1is exponen ial;
(ii) (low- ype IC binds) any bes eply o {H∗
1,H∗
1}yields he same payo as (σ,{H∗
0,
H∗
1});and
(iii) (agen s queue a all imes) pσ=0.
Agen s a e o e ed a choice be ween wo queues: one wi h a i s -come, i s -se ed
discipline and he o he wi h a andom-o de discipline. The agen s joining he queue
wi h a high belie , ha is, immedia ely a e ecei ing a posi i e lump-sum payo , a e
se ed in andom o de , he “ iskies ” discipline ha p o ides incen i es no o enege.
The agen s joining immedia ely a e ecei ing a nega i e lump-sum payo a e se ed
acco ding o a i s -come, i s -se ed queueing discipline; hence, hey a e exposed o
minimal isk.
4.2 The gene al case
When c>0, conside a ions abou queue leng h canno be igno ed. The ade-o be-
ween alloca i e e iciency and conges ion is mo e sub le, and pooling di e en ypes o
agen s by o e ing a single queue is some imes op imal.
Theo em 2. The e exis s a solu ion (σ,{H∗
0,H∗
1}) o he designe ’s p oblem (M). I is such
ha μH∗
1≥μH∗
1,δH∗
1≥δH∗
1, he capaci y cons ain (C) is binding, and one o he ollowing
holds:
(i) (pooling menu) H∗
0=H∗
1=H∗ o some H∗∈HNBUE,
(ii) (sepa a ing menu) H∗
0= H∗
1,and
(a) (FCFS/SIRO menu) H∗
0is degene a e and H∗
1is exponen ial;
(b) (low- ype IC binds) any bes eply o {H∗
1,H∗
1}yields he same payo as
(σ,{H∗
0,H∗
1});and
(c) (agen s queue a all imes) pσ=0.
The op imal menu can be o wo ypes: pooling o sepa a ing. In ui i ely, in he ab-
sence o mone a y ans e s, queueing is no only a byp oduc o sca ci y bu also se es
as a cos ly signaling de ice. A sc eening menu equi es agen s o engage in was e ul
queueing o signal hei ype and alloca es dedica ed capaci y o he “high ypes.” As in
Condo elli (2012), some imes he designe inds i op imal no o ex ac agen s’ p i a e
in o ma ion and ins ead o e s a single queue.
When he op imal menu is pooling, he op imal se ice discipline is ei he i s -
come, i s -se ed o se ice in andom o de wi h o wi hou a minimum wai ing- ime
equi emen . As shown below, bo h i s -come, i s -se ed and se ice-in- andom-
o de (wi h an a bi a y wai ing- ime equi emen >0) disciplines can eme ge as op i-
mal o some se s o pa ame e s (θ1,θ0,ρ0,ρ1,λ).
Theo e ical Economics 20 (2025) Queueing o lea n 641
A sepa a ing op imal menu coincides wi h he one in Theo em 1. The alue o he
in o ma ion acqui ed a each se ice is maximized: lea ning is so aluable ha agen s
queue a all imes, e en i queueing is cos ly. F om he pe spec i e o he indi idual
expe imen a ion p oblem, his does no mean ha explo ing is aluable a e e y belie .
Joining he queue has an op ion alue: i gua an ees he igh o be se ed a some poin
in he u u e when he belie will be highe . An agen joins he queue a a belie o 0
because he is ce ain ha he will no engage in explo a ion o some ime.
4.3 Discussion: Compa a i e s a ics
As men ioned be o e, when c>0, conside a ions abou queue leng h canno be igno ed.
In ac , he sepa a ing menu, i op imal, maximizes queue leng h. To he o he ex eme,
i he wai ing cos is high enough,16 he designe inds i op imal o o e a single i s -
come, i s -se ed queue, e en i his implies o going he possibili y o se e e u ning
agen s as soon as hey ejoin he queue.
Lemma 4. Fix any admissible se o pa ame e s (θ1,θ0,ρ0,ρ1,λ).
(i) The e exis s a csuch ha o c>c, nei he he sepa a ing menu no he pooling
se ice-in- andom o de queue is op imal.
(ii) I o e ing a single i s -come, i s -se ed queue is op imal, hen i minimizes he
a e age wai ing ime μ(equi alen ly, he queue leng h) among all easible disci-
plines.
Pa (i) o malizes he idea ha he bene i om se ing agen s who a e likely o
ha e a high p e ailing alua ion does no pay o o he inc eased conges ion when he
queueing cos is high, while pa (ii) lays ba e he ac ha he bene i om a i s -come,
i s -se ed queueing discipline comes om sho ening he queue.
The compa a i e s a ics wi h espec o λa e summa ized in Lemma 5and Figu e 3.
When he esou ce is sca ce, i is op imal o o e a single queue. A i s -come, i s -
se ed queue is subop imal o a high enough λ.
Lemma 5. Fix any admissible se o pa ame e s (θ1,θ0,ρ0,ρ1).
(i) The e exis s a λsuch ha o λ<λ, he sepa a ing menu is no op imal.
(ii) The e exis s a λsuch ha o e ing a single i s -come, i s -se ed queue is subop i-
mal o any λ≥λ.
The ole o pe sis ence is less clea -cu . On he one hand, he in o ma ional alue
om being se ed inc eases wi h pe sis ence, making sc eening mo e aluable. On he
o he hand, he cos o no se ing e u ning agen s as soon as hey join he queue de-
c eases wi h pe sis ence. In he ex eme case, i he s a e becomes a bi a ily pe sis en ,
16Because escaling (θ1,θ0,c)amoun s o escaling payo s bu does no a ec he implemen able se ,
inc easing cis equi alen o dec easing he gain om a ge ing he high ypes θ1−θ0.
642 Chia a Ma ga ia Theo e ical Economics 20 (2025)
Figu e 3. Compa a i e s a ics o (θ1,θ0,c)=(1, −3/4, 1/4)and ρ0=ρ1=ρ.Theshadingin-
dica es he ea u es o he op imal queueing discipline.
all non eneging single-queue se ice disciplines pe o m equally well. The ambiguous
impac o pe sis ence is shown in Figu e 3, whe e o simplici y, I se ρ0=ρ1=ρ.As
shown in he igh panel, o some se o pa ame e s, o e ing wo queues is op imal only
o an in e media e ange o he pe sis ence pa ame e ρ. While analy ical esul s a e
di icul o ob ain, nume al simula ions sugges ha he pa e ns iden i ied in Figu e 3
a e he ule.
Al hough he solu ion may be di e en o di e en objec i e unc ions, he analy-
sis p o ides he ools o s udy he queueing discipline ha minimizes wai ing ime o
maximizes he expec ed e u n om se ice.
In he wo king pape (Ma ga ia (2021)), I show ha he op imal menu can be i u-
ally implemen ed, in he sense ha i is possible o implemen an ou come a bi a ily
close o i wi h a single queueing discipline by aking ad an age o eneging.
5. Concluding ema ks
I s udy he op imal design o a queue o alloca e a esou ce o agen s wi h he e ogeneous
p e e ences. In his se up, a menu o sc een agen s akes he o m o mul iple queues (o
cus ome classes), wi h agen s being se ed in a di e en o de wi hin each o hem. The
op imal menu is (a mos ) bina y and has a simple s uc u e. When i is op imal o o e
wo dis inc queues, agen s a e se ed in a i s -come, i s -se ed manne in one queue
and in andom o de in he o he queue. When pooling is op imal, he single queue
is ei he i s -come, i s -se ed o se ice in andom o de , possibly wi h a minimum
wai ing- ime equi emen .
The analysis es s on h ee main assump ions: anonymi y, no ans e s, and lea n-
ing. I can be shown ha i any o he i s wo assump ions is d opped, i is possible o
es o e he i s bes , ha is, o achie e a o al payo a bi a ily close o λθ1.17 I agen s
17Each o hese ex ensions is examined in he wo king pape , Ma ga ia (2021).

Theo e ical Economics 20 (2025) Queueing o lea n 643
obse e he e olu ion o hei s a e, he anking o queueing disciplines is unambiguous:
he se ice-in- andom-o de discipline domina es any o he queueing discipline. I he
designe is able o de ec eneging, he op imal menu may in ol e a discipline ha ex-
poses he high- alua ion agen s o maximal a iabili y in wai ing ime by se ing a each
poin in ime some o he newly a i ed (as in a las -come, i s -se ed queue) and some
o he agen s who ha e been wai ing he longes (as in a i s -come, i s -se ed queue).
The model is s ylized in many espec s. Some assump ions a e o con enience.
Fo ins ance, he op imal menu is bina y e en i he alua ion akes mo e han wo al-
ues, p o ided ha he expec ed alua ion E[θi
|θi
=θj]e ol es mono onically o e ime.
The assump ion o a con inuum o agen s makes i possible o o mula e he bes - eply
p oblem as a simple Ma ko decision p oblem. A model wi h a ini e numbe o agen s
would allow o a ine analysis o he s a egic in e ac ion be ween hem, beyond he
gene al-equilib ium e ec cap u ed by he cu en model. Howe e , o he ex en ha
he p oblem educes o a “ wo-le el” op imiza ion p oblem, I belie e ha he main in-
sigh s would no be o e u ned in a se ing wi h a la ge bu ini e numbe o agen s.
Allowing o a iable capaci y would be use ul o s udy he wel a e implica ions o
conges ion in an en i onmen wi h luc ua ions.18 Mo e b oadly, because o he s a-
iona i y o he en i onmen and he independence assump ion, impo an aspec s o
queueing and lea ning ia expe imen a ion a e missing om he cu en model.19 Fo
example, co ela ion in agen s’ alua ions would in oduce he possibili y o obse a-
ional lea ning.
Appendix A: P oo s
A.1 P elimina ies
This sec ion con ains he o mal de ini ion o s a egies as impulse con ol policies. Fix
amenu{Hq:q∈Q}and le (Fi
) ≥0be he il a ion co esponding o he in o ma ion
o agen i. A s a egy o agen i,i∈[0, 1], is a sequence o andom imes and andom
a iables, (in e en ion imes and impulses a hese imes, espec i ely),
σ=(τk,ςk)∞
k=1,
whe e:
(i) 0 ≤τ1<τ
2<···,
(ii) o any k∈N,
τk=min >τ
k−1:dNi
>0,˜τk,
18In e es ingly, he pee - o-pee lending pla o m Zopa, a wo- ie queueing mechanism o alloca e
lende s’ unds o bo owing oppo uni ies, p io i izes e u ning lende s. Howe e , one may expec eco-
nomic luc ua ions o play a key ole in his en i onmen .
19Simila ly, he assump ion o a de e minis ic capaci y, allows o abs ac away om exogenous andom-
ness in he se ice ime, as would be he case i uni s o he good a i ed s ochas ically.
644 Chia a Ma ga ia Theo e ical Economics 20 (2025)
whe e ˜τkis a p edic able s opping ime adap ed o he il a ion (Fi
) ≥0(p e-
dic abili y en o ces he in o ma ional es ic ion ha , when queueing a ,an
agen chooses he s opping ime τa which he eneges condi ional on he e en
{in { > :dNi
>0}≥τ}),
(iii) o any k∈N,ςk∈ˆ
Qis a Fi
τk-measu able andom a iables, and
(i ) o any k∈N,i ςk=∅, henςk+1= ∅ a.s.
The s a egy de ines he ac ion p ocess (qi
,wi
) ≥0 aking alues in ˆ
Q×R,whe e
(qi
) ≥0is piecewise cons an , qi
τk=ςk,andwi
:=( −supτk≤ τk)1qi
=∅.
A.2 P oo s o Sec ion 3
I i s show ha he payo and he se ice a e om any s a iona y Ma ko s a egy
can be w i en as a unc ion o a ew su icien s a is ics and p o e he cha ac e iza-
ion o easible s a is ics in Lemma 3, as s a ed in Sec ion 3.3. Then I p o e he esul s in
Sec ion 3.1 and Sec ion 3.2.
A.2.1 P oo o Lemma 2The ansi ion ma ix o he semi-Ma ko chain in Figu e 1is
⎛
⎜
⎝
ρ1
ρ0+ρ1
+ρ0
ρ0+ρ1
ˆ
δσ
0
ρ0
ρ0+ρ11−ˆ
δσ
0
ρ1
ρ0+ρ11−ˆ
δσ
1ρ1
ρ0+ρ1
+ρ0
ρ0+ρ1
ˆ
δσ
1⎞
⎟
⎠.
The chain is posi i e ecu en and i educible. The unique s a iona y dis ibu ion is
1−m
m:=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ρ1
ρ0+ρ11−ˆ
δσ
1
ρ1
ρ0+ρ11−ˆ
δσ
1+ρ0
ρ0+ρ11−ˆ
δσ
0
ρ0
ρ0+ρ11−ˆ
δσ
0
ρ1
ρ0+ρ11−ˆ
δσ
1+ρ0
ρ0+ρ11−ˆ
δσ
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Fi s , assume νσ
s<∞,s=0, 1. Because any H∈Hhas no a oms a 0, he e exis ε>0
and ε>0, such ha P [Tn−Tn−1≤ε]≤1−ε. Mo eo e , he ansi ion ma ix is
unichain. Consequen ly, he long- un a e age payo can be compu ed using he e alua-
ion equa ions (by Pu e man (1994, Theo em 11.4.2, Chap e 11)). The long- un a e age
payo equals
mθ1−cˆμσ
1+(1−m)θ0−cˆμσ
0
mνσ
1+(1−m)νσ
0
=
ρ0
ρ0+ρ11−ˆ
δσ
0θ1−cˆμσ
1+ρ1
ρ0+ρ11−ˆ
δσ
1θ0−cˆμσ
0
ρ0
ρ0+ρ11−ˆ
δσ
0νσ
1+ρ1
ρ0+ρ11−ˆ
δσ
1νσ
0
. (10)
Theo e ical Economics 20 (2025) Queueing o lea n 645
The ime be ween Tnand Tn+1, o somen∈N, is independen o n.Hence, he
a e age numbe o upwa d jumps pe uni o ime con e ges almos su ely o he in e se
o he mean in e a i al ime (see, e.g., Asmussen (2003, Chap e V, P oposi ion 1.4)),
ha is, he se ice a e con e ges almos su ely o a cons an . F om di ec inspec ion o
(10), he a e a which he agen collec s lump sums con e ges almos su ely o
lim
→∞
1
N =1
mνσ
1+(1−m)νσ
0
=
ρ1
ρ0+ρ11−ˆ
δσ
1+ρ0
ρ0+ρ11−ˆ
δσ
0
ρ0
ρ0+ρ11−ˆ
δσ
0νσ
1+ρ1
ρ0+ρ11−ˆ
δσ
1νσ
0
. (11)
Because he game is symme ic and I analyze he s eady s a e, he long- un ac ion
o ime o which he agen is queueing coincides wi h he leng h o he queue in s eady
s a e and is equal o
mˆμσ,1 +(1−m)ˆμσ
0
mνσ
1+(1−m)νσ
0
=
ρ0
ρ0+ρ11−ˆ
δσ
0ˆμσ
1+ρ1
ρ0+ρ11−ˆ
δσ
1ˆμσ
0
ρ0
ρ0+ρ11−ˆ
δσ
0νσ
1+ρ1
ρ0+ρ11−ˆ
δσ
1νσ
0
.
Second, assume νσ
s=∞ o some s∈{0, 1}. In his case, he expec ed long- un a e -
age payo is ei he −∞ o 0, depending on
lim
T→∞ Eσ,{Hq}q∈QT
0
1q =∅ d ≧0.
I he p e ious limi is 0, he long- un a e age payo is 0; i di e ges o −∞ o he wise.
In bo h cases, he se ice a e con e ges almos su ely o ze o.
A.2.2 P oo o Lemma 3No ice ha he unc ion x→ e−(ρ0+ρ1)xis con ex. Thus, gi en
a mean μ>0, he minimum alue o he o he s a is ics is achie ed by he andom a i-
able ha is degene a e a μ. This, oge he wi h e−(ρ0+ρ1)x<1, p o es ha o any H∈H,
(δ,μ)∈. Fo he o he di ec ion, le (δ,μ)∈,δ= e−(ρ0+ρ1)μ. Conside a dis ibu ion
H ha andomizes be ween {ε/π,(μ−ε)/(1−π)}, wi h p obabili y (π,1−π),whe e
0<ε<μand π>0 a e chosen o sa is y
πe−(ρ0+ρ1)ε/π +(1−π)e−(ρ0+ρ1)(μ−ε)/(1−π)=δ.
I εwas 0, he p e ious equa ion would ha e a unique oo π∈(0, 1). Because he
le -hand side is con inuous in ε, he eexis anε>0andaπ∈(0, 1)such ha he
equali y is sa is ied. Clea ly, H∈H, and by cons uc ion, (δH,μH)=(δ,μ). I ins ead
δ=e−(ρ0+ρ1)μ, he s a is ics o he andom a iable degene a e a μa e (δ,μ).
To show he o he equi alence, le H∈HNBUE. Any NBUE andom a iable wi h
mean μis smalle han Exp[μ]in he con ex s ochas ic o de ,20 whe e Exp[μ]is he
20The andom a iable Xis said o be smalle han Yin he con ex o de i
Eφ(X)≤Eφ(Y) o all con ex unc ions φ:R→R,
p o ided he expec a ion exis s.
646 Chia a Ma ga ia Theo e ical Economics 20 (2025)
exponen ial andom a iable wi h he mean μ(see Shaked and Shan hikuma (2007,
Chap e 3, Theo em A.55)). Because he unc ion x→ e−(ρ0+ρ1)xis con ex, i ollows
ha any H∈HNBUE sa is ies
δH≤Ee−(ρ0+ρ1)Exp(μ)=1
1+(ρ0+ρ1)μ.
Consequen ly, any H∈HNBUE,(δH,μH)∈NBUE. To show he con e se, le (δ,μ)∈
NBUE. No e ha any degene a e dis ibu ion belongs o HNBUE.Fo μ>0, le D(μ)
deno e he andom a iable degene a e a μ. Then, by he p ope ies o he momen
gene a ing unc ion,
Ee−(ρ0+ρ1)(αD(μ)+(1−α)Exp(μ))=e−(ρ0+ρ1)αμ 1
1+(ρ0+ρ1)(1−α)μ.
Le αbe such ha
e−(ρ0+ρ1)αμ 1
1+(ρ0+ρ1)(1−α)μ=δ. (12)
Because δ∈[e−(ρ0+ρ1)μ,1
1+(ρ0+ρ1)μ]and he le -hand side is dec easing in α, o α∈
[0, 1], he e exis s a unique oo α o (12)in[0, 1]. This shows ha one can ind a andom
a iable H ha is a con olu ion o a degene a e dis ibu ion and an exponen ial dis i-
bu ion such ha (δ,μ)=(δH,μH). Because con olu ions o IHR dis ibu ions a e IHR,
he andom a iable αD(μ)+(1−α)Exp(μ)is IHR, and hence i is an NBUE andom
a iable. Because μ<∞, and nei he D(μ),no 0<Exp(μ)ha e a oms a 0, H∈HNBUE.
A.2.3 P oo o P oposi ion 1In he p oo , I assume ha {Hq}q∈Qis such ha he agen
has a bes eply ha yields s ic ly posi i e payo s. (In ligh o Lemma 11, his assump-
ion is wi hou loss o gene ali y.) As a gued, a s a egy σ∈can be desc ibed in e ms
o one s a e a iable only, he pos e io belie . I now in oduce some no a ion o de-
sc ibe s a iona y Ma ko s a egies in a way ha exploi s his ecu si i y. Fix a (pu e)
s a egy σ∈.Fo anyp∈[0, 1], de ine qσ:[0, 1]→ˆ
Q o be such ha , along he pa h
induced by σ,qσ(p )=q a.s. Addi ionally, de ine τσ:[0, 1]→R+so ha , a.s, along he
pa h induced by σ,
τσ(p):=in
τ≥0|w +τ−w +τ−= 0o q +τ= q −|p =p,dN +τ=0.
The maps τσand qσcomple ely cha ac e ize he s a egy σ∈: o any s a ing
belie pand ac ion (on he ecu en pa h induced by σ), he agen adjus s his ac ion
ei he a e an in e al o ime τσ(p), o when his belie jumps, whiche e occu s i s .
No e ha , he ime i akes o he belie o go om p o p, in he absence o jumps,
whene e ei he p<p
<ρ/
(ρ0+ρ1)o p>p
>ρ/
(ρ0+ρ1)is
ln(p)−ln p/(ρ0+ρ1)
whe e he unc ion :[0, 1]→Rwas de ined in (8).
Theo e ical Economics 20 (2025) Queueing o lea n 653
A.3 P oo s o Sec ion 4
A.3.1 P elimina ies F om he p oo o Lemma 2(see Sec ion A.2.1), hepayo oma
s a egy σ∈sa is ying he p ope ies in P oposi ion 1equals (wi h abuse o no a ion)
V(δˆ
Hσ
0,μˆ
Hσ
0,δˆ
Hσ
1,μˆ
Hσ
1,pσ),whe e
V(δ0,μ0,δ1,μ1,p)
:=(1−δ0)ρ0
ρ0+ρ1
+δ0p(θ1−μ1c)+(1−δ1)ρ1
ρ0+ρ1
(θ0−μ0c)
(1−δ0)ρ0
ρ0+ρ1
+δ0pμ1+(1−δ1)ρ1
ρ0+ρ1μ0+1
ρ0+ρ1
lnρ0
(1−p)ρ0−pρ1,
and he (a.s. limi o he) long- un se ice a e induced by σis equal o (wi h abuse o
no a ion) lim →∞ 1
N =S(δˆ
Hσ
0,μˆ
Hσ
0,δˆ
Hσ
1,μˆ
Hσ
1,pσ),whe e
S(δ0,μ0,δ1,μ1,p)
:=
δ0p+(1−δ0)ρ0
ρ0+ρ1
+(1−δ1)ρ1
ρ0+ρ1
δ0p+(1−δ0)ρ0
ρ0+ρ1μ1+(1−δ1)ρ1
ρ0+ρ1μ0+1
ρ0+ρ1
lnρ0
(1−p)ρ0−pρ1.
Fo la e pu poses, no ice ha he ollowing iden i ies hold ( he decomposi ion in (9)is
a special case o hese iden i ies):
V(δ0,μ0,δ1,μ1,p)
=S(δ0,μ0,δ1,μ1,p)m(δ0,δ1,p)(θ1−μ1c)
+1−m(δ0,δ1,p)(θ0−μ0c), (18)
S(δ0,μ0,δ1,μ1,p)=1
m(δ0,δ1,p)μ1+1−m(δ0,δ1,p)μ0+ (p), (19)
whe e
m(δ0,δ1,p):=
(1−δ0)ρ0
ρ0+ρ1
+δ0p
(1−δ0)ρ0
ρ0+ρ1
+δ0p+(1−δ1)ρ1
ρ1+ρ0
. (20)
The nex lemma shows ha o any uple o summa y s a is ics, (δ0,μ0,δ1,μ1), he e
exis s a unique op imal cu o s a egy wi hin he class o non eneging s a egies. Le
p∗:×→[0, ρ0/(ρ0+ρ1)] be de ined as
p∗(δ0,μ0,δ1,μ1)
:=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0i β(δ0,μ0,δ1,μ1)≤α(μ0,μ1)−1<−1,
ρ0
ρ0+ρ1
i β(δ0,μ0,δ1,μ1)≥0, o α(μ0,μ1)>0,
ρ0
ρ0+ρ1
×1−β(δ0,μ0,δ1,μ1)
W−1e−1+α(μ0,μ1)β(δ0,μ0,δ1,μ1)o he wise,

654 Chia a Ma ga ia Theo e ical Economics 20 (2025)
whe e W−1is he (nega i e b anch o he) Lambe unc ion and
α(μ0,μ1)=−
(ρ0+ρ1)θ1μ0−θ0μ1
θ1−cμ1
,
β(δ0,μ0,δ1,μ1)=−ρ0(θ1−cμ1)+(1−δ1)ρ1(θ0−cμ0)
δ0ρ0(θ1−cμ1).
(21)
The p oo o Lemma 10 is a ma e o edious algeb a and omi ed.
Lemma 10. Gi en (δ0,μ0,δ1,μ1)∈×, he e exis s a unique p∈[0, ρ0/(ρ0+ρ1)] ha
sol es maxp∈[0,ρ0/(ρ0+ρ1)] V(δ0,μ0,δ1,μ1,p). I equals p∗(δ0,μ0,δ1,μ1).
Le V∗(δ0,μ0,δ1,μ1):=V(δ0,μ0,δ1,μ1,p∗(δ0,μ0,δ1,μ1))and S∗(δ0,μ0,δ1,μ1):=
S(δ0,μ0,δ1,μ1,p∗(δ0,μ0,δ1,μ1)). I is easy o see ha he unc ion V∗is Ga eaux di -
e en iable in (δ0,μ0,δ1,μ1)whene e V∗(δ0,μ0,δ1,μ1)is s ic ly posi i e. A simila
ema k holds o p∗(δ0,μ0,δ1,μ1)whene e in e io , because i is a composi ion o
Ga eaux di e en iable unc ions. The ollowing ac assembles some echnical esul s o
be used la e .
Fac 1. Le (δ0,μ0,δ1,μ1)∈×and p∈[0, ρ0/(ρ0+ρ1))be such ha V(δ0,μ0,δ1,μ1,
p)>0.
(i) V(δ0,μ0,δ1,μ1,p)is s ic ly dec easing in μ0,μ1,andδ0, and s ic ly inc easing
in δ1.
(ii) p∗(δ0,μ0,δ1,μ1)is inc easing in μ0and μ1, inc easing in δ0(s ic ly i in e io ),
and dec easing in δ1.
Lemma 10 has an immedia e impo an consequence: he designe can always gua -
an ee ha he agg ega e payo is s ic ly posi i e by o e ing a single se ice-in- andom-
o de queue. The esul is o malized in he ollowing lemma whose p oo is omi ed.
Lemma 11. Fo any se o admissible pa ame e s (θ1,θ0,c,ρ0,ρ1,λ)∈R++ ×R++ ×R+×
R++ ×R++, he e exis s an equilib ium (σ,H)∈×H ha yields a s ic ly posi i e payo .
A.3.2 P oo o Theo em 2Whene e he cu o belie in Lemma 10 is in e io , i sa is ies
he i s -o de condi ion
V(δ0,μ0,δ1,μ1,p)=θ1−cμ1
μ1+ρ1
ρ0+ρ1
1−δ1
δ0
1
(1−p)ρ0−pρ1
. (22)
Fo con enience, le
κ(δ0,μ0,δ1,μ1)
:=1−δ1
δ0
1
1−p∗(δ0,μ0,δ1,μ1)ρ0−p∗(δ0,μ0,δ1,μ1)ρ1
, (23)
Theo e ical Economics 20 (2025) Queueing o lea n 655
so ha , when he op imal cu o is in e io , he ollowing iden i y holds:
mδ0,δ1,p∗(δ0,μ0,δ1,μ1)
=
1
1−δ1
ρ0
ρ0+ρ1
−1
(ρ0+ρ1)κ(δ0,μ0,δ1,μ1)
1
1−δ1
ρ0
ρ0+ρ1
+ρ1
ρ1+ρ0
−1
(ρ0+ρ1)κ(δ0,μ0,δ1,μ1)
. (24)
I shall e e o hese h ee equali ies se e al imes in he emainde o he p oo , in addi-
ion o he payo decomposi ion in (18)and oFac 1.
A.3.2.1 Relaxed p oblem I s a by sol ing he elaxed p og am
maxV∗δH0,μH0,δH1,μH1(RP)
o e (δH0,μH0)∈and (δH1,μH1)∈NBUE subjec o
V∗δH0,μH0,δH1,μH1≥V∗δH1,μH1,δH1,μH1,(IC-θ0)
V∗δH0,μH0,δH1,μH1≥V∗δH0,μH0,δH0,μH0,(IC-θ1)
S∗δH0,μH0,δH1,μH1≤λ.(C)
I i s s a e he solu ion o he p og am (RP), hen conclude he p oo o Theo em 2,and
inally p esen he p oo o he maximiza ion.
Lemma 12. The e exis s a solu ion o (RP). I is such ha (C) binds and ei he o he ol-
lowing holds:
(i) δH0=δH1and μH0=μH1;
(ii) δH0=e−(ρ0+ρ1)μH0,μH0>μ
H1,andδH1=1/(1+(ρ0+ρ1)μH1),and(IC-θ0)is
binding.
A.3.2.2 Conclusion o he p oo o Theo em 2I emains o p o e ha in he case o a
sepa a ing menu, agen s ha e no incen i es o enege. Since H0is degene a e, es a ing
is subop imal. The dis ibu ion H1is memo yless; hence, no agen can s ic ly bene i
om es a ing. Because μH1≤μH1and δH0<δ
H1, any agen wi h a belie abo e he
in a ian p obabili y p e e s H1 o H0; hus, e en along an a bi a ily long his o y wi h
no se ice, he high ype does no ind i op imal o lea e his queue and join he queue
H0.
A.3.2.3 P oo o Lemma 12 Fi s , I o mula e he domain es ic ions (δH0,μH0)∈
and (δH1,μH1)∈NBUE as explici cons ain s:
e−(ρ0+ρ1)μH1−δH1≤0, (WB-1)
e−(ρ0+ρ1)μH0−δH0≤0, (WB-0)
δH1−1
1+(ρ0+ρ1)μH1≤0. (EB-1)
656 Chia a Ma ga ia Theo e ical Economics 20 (2025)
De ine he Lag angian unc ion as
LδH0,μH0,δH1,μH1,η=V∗δH0,μH0,δH1,μH1+η1λ−S∗δH0,μH0,δH1,μH1
+η2V∗δH0,μH0,δH1,μH1−V∗δH1,μH1,δH1,μH1
+η3V∗δH0,μH0,δH1,μH1−V∗δH0,μH0,δH0,μH0
+η4δH0−e−(ρ0+ρ1)μH0+η51
1+(ρ0+ρ1)μH1−δH1
+η6δH1−e−(ρ0+ρ1)μH1,
whe e η∈R6
+is a ec o o mul iplie . I (δ∗
0,μ∗
0,δ∗
1,μ∗
1)∈(0, 1)×(0, ∞)×(0, 1)×(0, ∞)
and η∗≥0,η∗= 0a e such ha
(i) he cons ain s (IC-θ1), (IC-θ0), (C), (WB-1), (WB-0), (EB-1), and he complemen-
a y slackness condi ions a e sa is ied;
(ii) L(δH0
∗,μH0
∗,δH1
∗,μH1
∗,η∗)≥L(δH0,μH0,δH1,μH1,η∗), o any(δH0,μH0,δH1,
μH1)∈(0, 1)×(0, ∞)×(0, 1)×(0, ∞),
hen (δH0
∗,μH0
∗,δH1
∗,μH1
∗)is op imal. In he ollowing, I i s de i e quali a i e p ope ies
o any (δH0,μH0,δH1,μH1)∈(0, 1)×(0, ∞)×(0, 1)×(0, ∞)sa is ying (i) and (ii). I
is hen easy o show ha o any se o pa ame e s, such a pai exis s and he op imal
(δH0,μH0,δH1,μH1)mus sa is y he condi ions in he s a emen o Lemma 12.
Fi s , assume, h oughou he ollowing claims, ha (δH0,μH0,δH1,μH1)∈(0, 1)×
(0, ∞)×(0, 1)×(0, ∞).
Claim 1. I (δH0,μH0,δH1,μH1)sa is ies (IC-θ0)and(IC-θ1), δH0≤δH1.
P oo . By he a e age cos op imali y equa ions, he e exis s a unique (up o an addi-
i e cons an ) map u:{0, 1}→Rand a unique V∗∈Rsuch ha
u(0)=max
(δ,μ)−V∗μ−cμ +δp∗δH0,μH0,δH1,μH1+(1−δ)ρ0
ρ0+ρ1
×θ1+u(1)−θ0−u(0)
+θ0+u(0),
u(1)=max
(δ,μ)−V∗μ−cμ +δ+(1−δ)ρ0
ρ0+ρ1θ1+u(1)−θ0−u(0)+θ0+u(0),
whe e he maxima a e aken o e (δ,μ)∈{(δH0,μH0),(δH1,μH1)}. Clea ly, u(0)<u
(1).
I is easy o check ha , o (IC-θ0)and(IC-θ1) o be sa is ied, δH0≤δH1.
Claim 2. I (δH0,μH0,δH1,μH1)sa is ies (IC-θ0), δH0<δ
H1,μH1≤μH0,andS∗(δH0,
μH0,δH1,μH1)<S
∗(δH1,μH1,δH1,μH1), henp∗(δH0,μH0,δH1,μH1)=0.
Theo e ical Economics 20 (2025) Queueing o lea n 657
P oo . P oceeding by con adic ion, assume ha p∗(δH0,μH0,δH1,μH1)>0. Fi s , I
show ha his implies p∗(δH1,μH1,δH1,μH1)>0. In ac , because he igh -hand side o
(22) is s ic ly inc easing in δH0and s ic ly dec easing in p, o anyh=(hδH0,hμH0,0,0
),
hδH0>0andhμ0≤0such ha ∇V∗(δH0,μH0,δH1,μH1)·h≤0, ∇p∗(δH0,μH0,δH1,μH1)·
h>0. As a esul , (IC-θ0) implies ha i p∗(δH0,μH0,δH1,μH1)>0, p∗(δH1,μH1,δH1,
μH1)>0.
Second, by (22), (IC-θ0)implies
κδH0,μH0,δH1,μH1≤κδH1,μH1,δH1,μH1
mδH0,δH1,p∗δH0,μH0,δH1,μH1≤mδH1,δH1,p∗δH1,μH1,δH1,μH1.
Since μH1≤μH0, i ollows ha S∗(δH1,μH1,δH1,μH1)≤S∗(δH0,μH0,δH1,μH1),acon-
adic ion.
Claim 3. I (δH0,μH0,δH1,μH1)sol es (RP)andμH0<μ
H1, henp∗(δH0,μH0,δH1,
μH1)=0.
P oo . Fi s , no ice ha μH0<μ
H1and (IC-θ1) imply ha (IC-θ0) is slack. Assume ha
p∗(δH0,μH0,δH1,μH1)>0, so ha he i s -o de condi ions hold. Conside a change
along a di ec ion h=(0, hμH0,0,hμH1),hμ0>0, hμ1<0such ha
mδH0,δH1,p∗δH0,μH0,δH1,μH1hμH1
+1−mδH0,δH1,p∗δH0,μH0,δ1,μH1hμH0=0
I ∇V∗(δH0,μH0,δH1,μH1)·h≤0, (22)implies∇κ(δH0,μH0,δH1,μH1)·h>0and
∇p(δH0,μH0,δH1,μH1)·h>0. Hence, ∇S∗(δH0,μH0,δH1,μH1)·h<0. As a esul , one
can ind h
μH1such ha
∇S∗δH0,μH0,δH1,μH1·0, hμH0,0,h
μH1=0,
∇V∗δH0,μH0,δH1,μH1·0, hμH0,0,h
μH1>0.
I ins ead ∇S∗(δH0,μH0,δH1,μH1)·h>0, le h
μH0>h
μH0be such ha ∇S∗(δH0,μH0,
δH1,μH1)·(0, h
μH0,0,hμH1)=0.By he i s pa o hep oo ,∇V∗(δH0,μH0,δH1,μH1)·
h>0, con adic ing he op imali y o (δH0,μH0,δH1,μH1).
Claim 4. I (δH0,μH0,δH1,μH1)sol es (RP), (δH0,μH0)= (δH1,μH1),andp∗(δH0,μH0,
δH1,μH1)=0, hen(WB-0)and(EB-1)a ebinding.
P oo . Suppose i s ha (WB-0) is slack. In his case,
∂S∗δH0,μH0,δH1,μH1
∂δ0
=− μH0−μH1ρ0
1−δH0ρ0+1−δH1ρ1
∂S∗δH0,μH0,δH1,μH1
∂μ0
, (25)
658 Chia a Ma ga ia Theo e ical Economics 20 (2025)
and
∂V ∗δH0,μH0,δH1,μH1
∂δ0
=− μH0θ1−μH1θ0ρ0
1−δH0ρ0θ1+1−δH1ρ1θ0
∂V ∗δH0,μH0,δH1,μH1
∂μ0
. (26)
Because θ0<0<θ
1,21
μH0−μH1ρ0
1−δH0ρ0+1−δH1ρ1
<μH0θ1−μH1θ0ρ0
1−δH0ρ0θ1+1−δH1ρ1θ0
. (27)
Since
∂V ∗δH0,μH0,δH1,μH1
∂μ0
<0, ∂S∗δH0,μH0,δH1,μH1
∂μ0
<0,
he e exis s a di ec ion h=(hδH0,hμH0,0,0
),hδH0<0, hμ0>0, along which
∇S∗δH0,μH0,δH1,μH1·h≤0, ∇V∗δH0,μH0,δH1,μH1·h>0.
Because along he di ec ion hall o he cons ain s a e ei he unchanged o elaxed, a
he op imum, (WB-0) mus bind. Assume nex ha (EB-1)isslack.Since,
∂S∗δH0,μH0,δH1,μH1
∂δ1
=μH0−μH1ρ1
1−δH0ρ0+1−δH1ρ1
∂S∗δH0,μH0,δH1,μH1
∂μ1
,
∂V ∗δH0,μH0,δH1,μH1
∂δ1
=μH0θ1−μH1θ0ρ1
1−δH0ρ0θ1+1−δH1ρ1θ0
∂V ∗δH0,μH0,δH1,μH1)
∂μ1
,
by (27), he e exis s a di ec ion h=(0, 0, hδH1,hμH1),hδH1>0andhμ1>0, along which
all cons ain s a e elaxed o unchanged and he objec i e unc ion is inc eased. Hence,
(EB-1) mus bind a he op imum.
Claim 5. I (δH0,μH0,δH1,μH1)sol es (RP), p∗(δH0,μH0,δH1,μH1)=0,and(WB-1)is
slack, (IC-θ0)binds.
21In ligh o he de ini ion o G(δH0,μH0,δH1,μH1,p), o any candida e op imal uple
(δH0,μH0,δH1,μH1), he denomina o o he e m on he igh -hand side o equa ion (27) is s ic ly
posi i e. Fo o he wise, he agg ega e payo s would be s ic ly nega i e, o any s a egy ha p esc ibes
queueing wi h posi i e p obabili y.

Theo e ical Economics 20 (2025) Queueing o lea n 659
P oo . Assume by con adic ion ha (IC-θ0)isslack.Le h=(hδH0,hμH0,0,0
),hδH0<
0, hμ0>0beadi ec ionsuch ha
22
hδH0+(ρ0+ρ1)e−(ρ0+ρ1)μ0hμ0=0. (28)
Assume i s ha ∇V∗(δH0,μH0,δH1,μH1)·h<0. Then, by (25)–(27), ∇S∗(δH0,μH0,δH1,
μH1)·h<0. Le hen hμH1<0besuch ha ∇S∗(δH0,μH0,δH1,μH1)·(hδH0,hμH0,0,
hμH1)=0. Because ∂m(δH0,δH1,p)/∂δH0<0, i ∇S∗(δH0,μH0,δH1,μH1)·h=0,
∇V∗(δH0,μH0,δH1,μH1)·h>0. As no cons ain is iola ed along ha di ec ion, his
con adic s he op imali y o (δH0,μH0,δH1,μH1). Assume nex ha ∇V∗(δH0,μH0,δH1,
μH1)·h>0. Again, by (25)–(27), i is possible o ind a di ec ion h=(h
δH0,h
μH0,0,0
)
ha does no iola e (WB-0)andsuch ha ∇S∗(δH0,μH0,δH1,μH1)·h=0and∇V∗(δH0,
μH0,δH1,μH1)·h>0. As no cons ain is iola ed along ha di ec ion, his con adic s
he op imali y o (δH0,μH0,δH1,μH1).
Claim 2and Claim 3imply ha a anyop imal(δH0,μH0,δH1,μH1),(δH0,μH0)=
(δH1,μH1),p∗(δH0,μH0,δH1,μH1)=0. Hence, by Claim 4, a any op imal such a
(δH0,μH0,δH1,μH1),(WB-0)and(EB-1) a e binding. As when (EB-1) is binding, (WB-1)
is slack, Claim 5implies ha (IC-θ0) binds, which implies μH0>μ
H1. I emains o p o e
ha (C) binds a any such op imal (δH0,μH0,δH1,μH1).
Claim 6. I (δH0,μH0,δH1,μH1)is op imal, p∗(δH0,μH0,δH1,μH1)=0,δH0<δ
H1,and
(IC-θ0) binds, he cons ain (C)binds.
P oo . Conside a di ec ion h=(hδH0,hμH0,0,0
),hδH0>0, and hμ0<0such ha
hδH0+(ρ0+ρ1)δH0hμ0=0. I can be checked ha
∂V ∗δH0,μH0,δH1,μH1
∂δ0
=μH0θ1−μH1θ0ρ0
1−δH0ρ0θ1+1−δH1ρ1θ0
∂V ∗δH0,μH0,δH1,μH1
∂μ0
<0.
Since p∗(δH0,μH0,δH1,μH1)=0,
μH0θ1−μH1θ0ρ0
1−δH0ρ0θ1+1−δH1ρ1θ0
<1
(ρ0+ρ1)δH0.
As a esul , ∇V∗(δH0,μH0,δH1,μH1)·h>0. I (C) does no bind, no cons ain is iola ed
along he di ec ion h, con adic ing he op imali y o (δH0,μH0,δH1,δH1).
Claim 7. I (δ,μ,δ,μ)is op imal, (C)binds.
22The es ic ion (28) akesca eo (WB-0).
660 Chia a Ma ga ia Theo e ical Economics 20 (2025)
P oo . Assume (C)doesno bind.I e−(ρ0+ρ1)μ<δ,so ha (WB-1)and(WB-0)a eslack,
conside a di ec ion h=(0, hμ,0,hμ),hμ<0. The ac ha ∇V∗(δ,μ,δ,μ)·h>0con-
adic s he op imali y o (δ,μ,δ,μ). Conside nex he case in which δ=e−(ρ0+ρ1)μand
conside a di ec ion h=(hδ,0,hδ,0
),hδ>0. I is e i ied ha
∂mδ,δ,p∗(δ,μ,δ,μ)
∂δ0
+∂mδ,δ,p∗(δ,μ,δ,μ)
∂δ1
>0
∂δ,μ,δ,μ,p∗(δ,μ,δ,μ)
∂δ0
+∂δ,μ,δ,μ,p∗(δ,μ,δ,μ)
∂δ1
>0.
Since he change p∗(δ,μ,δ,μ)can be neglec ed (as ollows om he en elope heo em),
∇V∗(δ,μ,δ,μ)·h>0, yielding he desi ed con adic ion.
A.3.3 P oo o Theo em 1Fi s , I show when c=0, se ice-in- andom-o de is he bes
discipline when he designe is cons ained o o e a single non eneging queueing dis-
cipline. Second, I show ha he e exis a menu ha ou pe o ms he bes se ice-in-
andom-o de discipline.
Le (δ,μ)∈NBUE and h=(hδ,hμ,hδ,hμ)∈R4
++ be a di ec ion such ha ∇V∗(δ,μ,
δ,μ)·h=0. The e a e wo cases. I ρ0θ1+ρ1θ0>0, he op imal cu o p( ,μ,δ,μ)
is s ic ly inc easing in δand μ. Since o ixed p,∇m(δ,δ,p)·h>0andm(δ,δ,p)is
s ic ly inc easing in p,
∇mδ,δ,p∗(δ,μ,δ,μ)·h+∂mδ,δ,p∗(δ,μ,δ,μ)
∂p ∇p∗(δ,μ,δ,μ)·h>0. (29)
By (18), ∇V∗(δ,μ,δ,μ)·h=0onlyi ∇S∗(δ,μ,δ,μ)·h<0. Nex , I show ha e en
i ρ0θ1+ρ1θ0≤0, (29) holds. To do so, I ew i e he agen ’s bes eply p oblem in
Lemma 10 as a choice o e mins ead o a choice o e op imal cu o s p.Tha is,gi en
(δ,μ)∈, he e exis s a unique m∈(0, 1) ha sol es
max
m∈[ρ0/(ρ0+ρ1),ρ0/(ρ0+(1−δ)ρ1))
mθ1+(1−m)θ0
μ+(1−m)1
ρ0+ρ1
ln(1−m)δρ0
(1−m)ρ0−(1−δ)mρ1
The i s -o de condi ions ead
(1−δ)mθ1+(1−m)θ0ρ1
ρ0+ρ1
−(θ1−θ0)μ+θ1
ρ0+ρ1
ln(1−m)δρ0
(1−m)ρ0−(1−δ)mρ1(1−m)ρ0−(1−δ)mρ1=0.
I can be shown ha he le -hand side he equa ion abo e is dec easing in δand
μ, and, using he assump ion ρ0θ1+ρ1θ0≤0, i is inc easing in m.Hence,by heim-
plici unc ion heo em, along a di ec ion h=(hδ,hμ,hδ,hμ)∈R4
++,(29)mus hold,and
again, by (18), ∇V∗(δ,μ,δ,μ)·h=0onlyi ∇S∗(δ,μ,δ,μ)·h<0.
As a esul , he op imal pai (δ,μ)∈NBUE mus lie a he eas bounda y o he se
NBUE. O he wise, one could inc ease wel a e by inc easing δand μwi hou iola ing
he capaci y cons ain .
Theo e ical Economics 20 (2025) Queueing o lea n 661
To ind he i s -come i s -se ed/se ice-in- andom-o de menu ha ou pe o ms
he bes single se ice-in- andom-o de queue, I show ha he ollowing sys em, which
iden i ies a candida e op imal menu, has a solu ion:
(1−δ0)ρ0
ρ0+ρ1
θ1+(1−δ1)ρ1
ρ0+ρ1
θ0
(1−δ0)ρ0
ρ0+ρ1
1−δ1
(ρ0+ρ1)δ1
+(1−δ1)ρ1
ρ0+ρ1
ln(1/δ0)
ρ0+ρ1
=δ1θ1
1−δ1
ρ0+ρ1
+ρ1
ρ0+ρ1
1−δ1
ρ0−(ρ0+ρ1)p∗1−δ1
(ρ0+ρ1)δ1
,δ1,1−δ1
(ρ0+ρ1)δ1
,δ1
, (30)
(1−δ0)ρ0
ρ0+ρ1
+(1−δ1)ρ1
ρ0+ρ1
(1−δ0)ρ0
ρ0+ρ1
1−δ1
(ρ0+ρ1)δ1
+(1−δ1)ρ1
ρ0+ρ1
ln(1/δ0)
ρ0+ρ1
=λ, (31)
0≤(1−δ0)ρ0
ρ0+ρ1
θ1+(1−δ1)ρ1
ρ0+ρ1
θ0−ρ0δ0θ1
ln(1/δ0)
ρ0+ρ1
−θ0
1−δ1
(ρ0+ρ1)δ1. (32)
Fo a ixed δ0∈(0, 1), he le -hand side o (31) is inc easing in δ1, ends o in ini y
as δ1→1, and o 0 as δ1→0. As a esul , he e exis s a con inuous cu e C⊂(0, 1)2, he
i s and second coo dina es co esponding o δ0and δ1, espec i ely, ha is a solu ion
o (31). I is easy o see ha {(0, 1),(1, 0)}⊂C.
Fo a ixed δ1∈(0, 1), he le -hand side o (32) is dec easing in δ0;le D0∈(0, 1)2
be he se o poin s ha sa is y (32)asequali y. Deno ebyD0 he se o poin s lying
on o abo e he cu e D0. The le -hand side o (30) is inc easing in δ0i and only i
(δ0,δ1)∈D0. Hence, he e exis s a con inuous cu e D⊂D0 ha sol es (30). Because
{(0, 0),(1, 1)}⊂D, by he in e media e alue heo em, he wo cu es Dand Cc oss and
(δ0,δ1)∈C∩D⊂D0sol es he sys em o (30)–(32).
Any solu ion o he sys em desc ibes an incen i e-compa ible and easible menu
such ha agen s ha e incen i e o join he i s -come i s -se ed queue as soon as hei
belie jumps o ze o. By Lemma 13, o any(δ0,δ1)∈C∩D,
mδ0,δ1,p∗δ1,1−δ1
(ρ0+ρ1)δ1
,δ1,1−δ1
(ρ0+ρ1)δ1<m
(δ0,δ1,0
),
which implies ha
S∗δ0,ln(1/δ0)
ρ0+ρ1
,δ1,1−δ1
(ρ0+ρ1)δ1<S
∗δ1,1−δ1
(ρ0+ρ1)δ1
,δ1,1−δ1
(ρ0+ρ1)δ1.
Tha is, he se ice-in- andom-o de discipline which yields he same payo as a candi-
da e menu is un easible. Consequen ly, he bes easible se ice-in- andom-o de dis-
cipline is ou pe o med by any candida e menu ha sol es (30)–(32).
Lemma 13. Suppose δ0∈(0, 1)and δ1∈(0, 1)sol es (30)–(32)andV∗(δ,ln(1/δ0)/(ρ0+
ρ1),δ1,1−δ1/((ρ0+ρ1)δ1)) >0.Then
m∗δ1,δ1,p∗δ1,1−δ1
(ρ0+ρ1)δ1
,δ1,1−δ1
(ρ0+ρ1)δ1<m
(δ0,δ1,0
).
662 Chia a Ma ga ia Theo e ical Economics 20 (2025)
P oo . One can check ha o any δ1∈(0, 1),p∗(δ1,1−δ1/((ρ0+ρ1)δ1),δ1,1−δ1/
((ρ0+ρ1)δ1))>0 and, by assump ion, p∗(1−δ1/((ρ0+ρ1)δ1),δ1,1−δ1/((ρ0+ρ1)δ1),
δ1)<ρ
0/(ρ0+ρ1), so he i s -o de condi ion holds. Hence,
δ0·ρ0
ρ0+ρ1
<δ
1ρ0
ρ0+ρ1
−p∗1−δ1
(ρ0+ρ1)δ1
,δ1,1−δ1
(ρ0+ρ1)δ1
,δ1
The esul ollows om he de ini ion o m(δ0,δ1,p).
A.3.4 P oo o Lemma 4
A.3.4.1 P oo o (i) Fi s , when he candida e menu is op imal, he o al payo is
bounded abo e by λθ1−c, which is nega i e o c>λθ
1. Howe e , as shown a e Fac 1,
he designe can always gua an ee ha he agg ega e payo is s ic ly posi i e by o -
e ing a single se ice-in- andom-o de queue. Hence, he sepa a ing menu canno be
op imal o su icien ly high c.
Second, I show ha o su icien ly high c, se ice-in- andom-o de is no op imal
e en i he designe is cons ained o a single non eneging queue.
Le (δSIRO
c,μSIRO
c)be he s a is ics o he bes easible se ice-in- andom-o de
queue when he queueing cos is c.Tha is,
S∗δSIRO
c,μSIRO
c,δSIRO
c,μSIRO
c=λ,μSIRO
c=1−δSIRO
c
(ρ0+ρ1)δSIRO
c
.
(By Lemma 11, his sys em has a solu ion o any c>0.)
Claim 8. The ollowing hold:
(i) limc→∞ δSIRO
c=1;
(ii) limc→∞ p∗(δSIRO
c,μSIRO
c,δSIRO
c,μSIRO
c)=ρ0/(ρ0+ρ1).
P oo .Fi s ,(δ,μ,δ,μ,p)is s ic ly dec easing in pwhen e alua ed a p∗(δ,μ)and
δ,(1−δ)/(ρ0+ρ1)δ,δ,(1−δ)/(ρ0+ρ1)δ,p
is s ic ly inc easing in δ. Second, o a ixed (δ,μ), he unc ionp∗(δ,μ,δ,μ)is inc eas-
ing in cand
p∗δ,(1−δ)/(ρ0+ρ1)δ,δ,(1−δ)/(ρ0+ρ1)δ
is dec easing in δ, in bo h cases s ic ly i he cu o belie is in e io . Addi ionally,
p∗(δSIRO
c,μSIRO
c,δSIRO
c,μSIRO
c)>0 (see p oo o Lemma 11). Hence, (i) ollows by he im-
plici unc ion heo em. To show (ii), no ice ha as c→∞,S∗(δSIRO
c,μSIRO
c,δSIRO
c,μSIRO
c)
is bounded only i −ln (p∗(δSIRO
c,μSIRO
c,δSIRO
c,μSIRO
c)) →∞, ha is,i p∗(δSIRO
c,μSIRO
c,
δSIRO
c,μSIRO
c)→ρ0/(ρ0+ρ1).