Games with incomplete information A simplified exposition with inventory management applications

16 Pages·2011·0.8 MB·English

by  H. Wu

Int.J.ProductionEconomics133(2011)562–577
ContentslistsavailableatScienceDirect
Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
Review
Games with incomplete information: A simplified exposition with inventory
management applications
H. Wu, M. Parlar(cid:2),1
DeGrooteSchoolofBusiness,McMasterUniversity,Hamilton,Ontario,CanadaL8S4M4
a r t i c l e i n f o a b s t r a c t
Articlehistory: Inmostexistingliteratureinsupplychainmanagementitisassumedthattheplayerspossesscomplete
Received31August2010 informationaboutthegame,i.e.,theplayers’payoff(objective)functionsareassumedtobecommon
Accepted8June2011 knowledge. For static and dynamic games with complete information, the Nash equilibrium and
Availableonline21June2011
subgameperfectequilibriumarethestandardsolutionconcepts,respectively.Forstaticanddynamic
Keywords: gameswithincompleteinformation,theBayesianNashequilibriumandperfectBayesianequilibrium,
Gametheory respectively,areusedassolutionconcepts.Afterpresentingabriefreviewofthestaticanddynamic
Incompleteinformation gamesundercompleteinformation,theapplicationofthesetwogamesininventorymanagementis
Inventorytheory illustratedby using asingle-period stochastic inventoryproblem with two competing newsvendors.
Next,weillustratetheBayesianNashandperfectBayesianequilibriumsolutionconceptsforthestatic
anddynamicgamesunderincompleteinformationwithtwocompetingnewsvendors.Theexpository
natureofourpapermayhelpresearchersininventory/supplychainmanagementgaineasyaccessto
thecomplicatednotionsrelatedtothegamesplayedunderincompleteinformation.
&2011ElsevierB.V.Allrightsreserved.
Contents
1. Introduction......................................................................................................563
2. Gameswithcompleteinformation....................................................................................564
2.1. ModelI:staticgameswithcompleteinformation..................................................................564
2.1.1. Discretestrategies ....................................................................................564
2.1.2. Continuousstrategies..................................................................................565
2.2. ModelII:dynamicgameswithcompleteinformation...............................................................566
2.2.1. Discretestrategies ....................................................................................566
2.2.2. Continuousstrategies..................................................................................567
3. ModelIII:staticgameswithincompleteinformation .....................................................................567
3.1. Discretestrategies...........................................................................................568
3.2. Continuousstrategies.........................................................................................570
3.2.1. PlayerP1hastwotypesandplayerP2hasonetype.........................................................570
3.2.2. PlayersP1andP2bothhavetwotypes ...................................................................571
4. ModelIV:dynamicgameswithincompleteinformation...................................................................572
4.1. Discretestrategies...........................................................................................572
4.1.1. Strategicform........................................................................................572
4.1.2. Computingtheequilibriaintheextensiveform.............................................................573
4.1.3. Intuitivecriterion.....................................................................................574
4.2. Continuousstrategies.........................................................................................574
(cid:2)Correspondingauthor.
E-mailaddress:[email protected](M.Parlar).
1ResearchsupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada.
0925-5273/$-seefrontmatter&2011ElsevierB.V.Allrightsreserved.
doi:10.1016/j.ijpe.2011.06.004
H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577 563
5. Conclusion.......................................................................................................576
Acknowledgments.................................................................................................576
References .......................................................................................................576
1. Introduction
problemsdealingwithgamesundercompleteinformation.Cachon
andNetessine(2004)brieflymentionsignaling,screeningandthe
Gametheorystudiesmultiple-persondecisionproblemsinvol- Bayesian games where the games are played under incomplete
ving conflict or cooperation. Following the publication of von information, i.e., at least one of the players does not know the
Neumann and Morgenstern’s (1944) seminal book, interest in other players’ objective function. As examples of games played
potential applications of game theory reached a peak in the under incomplete information, they cite Cachon and Lariviere
following decade. The fundamental solution concepts of game (2001) who applied a signaling game to study a contracting
theory (e.g., the Nash equilibrium for non-cooperative games, problem with information sharing in a one-supplier, one-manu-
Nash (1950), and the Shapley value for cooperative games, facturer supply chain, and Cachon and Lariviere (1999) who
Shapley (1953)) were developed in the 1950s and were used to studied a capacity allocation problem with information sharing
analyze problems in diverse areas including economics, political issue between a supplier and several downstream retailers.
science,management-laborarbitration,philosophyandwarfare. Contract design problems also involve games of incomplete
Early applications of game theory considered games of ‘‘com- information;one of theearliest papersinthis areais by Corbett
pleteinformation’’,whereeachplayer’spayoff(objective)function (2001) who applied the principal-agent theory to design an
iscommonknowledgeforallplayers.However,thestringent(and inventory contract in the context of the (Q,r) model. In a more
unrealistic) assumption of complete information becamea barrier recentpaper,ChuandLee(2006)studiedaninformationsharing
to successful implementation of game theoretic ideas because in problem in a vertical supply chain with one vendor and one
mostcompetitiveproblemstheplayersarenotprivytoeachother’s retailer and employed the perfect Bayesian equilibrium as the
payoff functions. For example, two firms competing for the same solutionconceptusedindynamicgamesplayedunderincomplete
marketdemanddonothavecompleteinformationoneachother’s information. In a recent paper, Costantino and DiGravio (2009)
production cost functions. Similarly, in a sealed-bid auction, the combineconceptsfromgametheoryandfuzzylogictoanalyzea
bidders do not know each other’s valuations. In the late 1960s, bargainingproblemwithincompleteinformation.
Harsanyi (1967) developed solution concepts for games with The reviews by Cachon and Netessine (2004) and Leng and
incomplete, i.e., asymmetric, information (also known as the Parlar(2005)revealthatthereisapaucityofpapersthatdealwith
Bayesian games). In such games, players’ payoff functions are no games played under incomplete information. However, in recent
longercommonknowledge;instead,atleastoneplayerisuncertain years publications have begun to appear that analyze games
aboutanotherplayer’spayofffunction.WithHarsanyi’sdiscoveryof played under incomplete information. Since most realistic SCM
the new solution concepts for incomplete information games, problemsinvolvecompetitiveinteractionswithincompleteinfor-
interest in game theory was heightened in the last two decades mation,itwouldbeusefultoprovideanexpositionofsuchgames
andgametheoryonceagainbecameanimportanttoolthatcanbe with applications to a specific area in SCM, namely, inventory
usedtoanalyzerealisticproblemsofcompetitivesituations. management.Withthisinmind,wewrotethispapertopresenta
Operationsresearcherswereearlyusersofgametheoryascan simplifiedtreatmentofgameswithincompleteinformationwith
beseenintheoperationsresearchtextspublishedinthe1950sand applicationsinstochasticinventorymanagement.
1960s. The textbooks by Churchman et al. (1958), Sasieni et al. We follow the same framework as in Gibbons (1992, 1997)
(1959), Hillier and Lieberman (1986) and Ackoff and Sasieni whohasalsoconsideredstaticanddynamiccompleteandincom-
(1968), all include a chapter on competitive problems. All four plete information games and their applications in economics.
textscoverzero-sumgamesandall,exceptHillierandLieberman Gibbons’sclassificationresultsinfourcategories:(i)staticgames
(1986),presentafewexamplesofnon-zerosumgamesinvolving withcompleteinformation(forwhichthesolutionconceptused
bidding strategies. Shubik (1955) reviewed early publications in is the Nash equilibrium), (ii) dynamic games with complete
this area. However, after the initial excitement generated by its information (subgame perfection and the Stackelberg equili-
potential applications, operations researchers’ interest in game brium), (iii) static games with incomplete information (the
theory seemed to have waned during the 1970s and the 1980s. BayesianNashequilibrium),and(iv)dynamicgamewithincom-
But the last two decades have witnessed a renewed interest by plete information (perfect Bayesian Nash equilibrium). We start
academicsand practitioners inthe management ofsupply chains by briefly describing static and dynamic complete information
and a new emphasis on the interactions among decision makers games.Thisisfollowedbyamoredetailedexpositionofstaticand
(‘‘players’’) constituting a supply chain. This has resulted in the dynamicincompleteinformationgames.Wefirstillustrateeachof
proliferation of game theoretical publications in operations the four cases (which we call ‘‘Models’’) with a simple discrete
research/management science/operations management (OR/MS/ gamewhere eachplayer has two moves.Foreach case, we then
OM)journalsdealingwiththe use ofgametheory inthe compe- present a single-period stochastic inventory game with two
titive and cooperative problems arising in supply chain manage- competing newsvendors with the players’ decision variables as
ment(SCM).Foranexcellentreviewofgametheoreticapplications continuousvalues.Whileourpaperisexpositoryinnature,italso
in supply chain management we refer the reader to Cachon and contributes to the literature by presenting explicit methods for
Netessine(2004);seealsoamorerecentreviewbyLengandParlar dealing with static and dynamic inventory games under incom-
(2005). plete information and computing the Bayesian Nash and perfect
IntheirrespectivereviewsofgametheoryapplicationsinSCM, Bayesian equilibrium for such games. For another, and very
CachonandNetessine(2004)andLengandParlar(2005)eachcite detailed, treatment of static and dynamic complete and incom-
more than 100 papers. It is interesting to note that a large pleteinformationgamesandtheirapplicationsineconomics;see,
majority of the reviewed papers make the simplified (and Fudenberg and Tirole (1992). Myerson (1991) and Osborne and
frequently unrealistic) assumption that all players know each Rubinstein (1994) also provide excellent overviews of game
other’sobjectivefunctionswithcertainty.Thatis,theyinvestigate theorywitheconomicapplications.
564 H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577
In Section 2, we briefly review the well-studied games of by his order quantity q and also by P2’s order quantity q .
1 2
complete information and discuss the solution concepts of the (Clearly, if P1 chooses a low value of q , this may result in
1
Nashequilibrium(ModelI,forstaticgames)andsubgameperfect shortagesforhim and thus P2maybenefit fromthis assomeof
equilibrium (Model II, for dynamic games). In Section 3, we P1’s unsatisfied customers may switch to P2.) Similarly, P2’s
presentadiscussionofastaticgameunderincompleteinforma- expected profit J is affected by her order quantity q and also
2 2
tion and discuss the solution concept of the Bayesian Nash byP1’sorderquantityq .Thus,wewriteJ ðq ,q ÞandJ ðq ,q Þas
1 1 1 2 2 1 2
equilibrium.InSection4,wediscussthecaseofdynamicgames theexpectedprofitsofplayersP1andP2,respectively.
under incomplete information and use the solution concept of
perfect Bayesian equilibrium to solve the game. Section 5 con-
cludesthepaperwithabriefsummary. 2.1.1. Discretestrategies
Since our inventory applications are concerned with games Considerfirstasimplesituationwherethenewsvendors’order
played by two newsvendors, we assign the male gender to the quantitiesarelimitedtotakeonlyoneoftwopossiblevalues,say
first newsvendor and the female gender to the second news- low or high. Thus, P1 chooses either L (low) or H (high), i.e.,
1 1
vendor in order to minimize the confusion that may arise when q AfL ,H g, and P2 chooses either ‘ (low) or h (high), i.e.,
1 1 1 2 2
werefertotheplayers. q Af‘ ,h g. For each combination of order quantities, the news-
2 2 2
vendors’ expected profits are given in strategic form(or, normal
form)inTable1asapairofnumbersðJ ,J Þ.(Ingeneral,theðJ ,J Þ
1 2 1 2
2. Gameswithcompleteinformation values represent the players’ expected utilities (Luce and Raiffa,
1957;Straffin,1993,Chapter9;vonNeumannandMorgenstern,
Inthissectionwepresentasummaryofgamesplayedunder 1944,Chapter3),butinthispaperweassumethattheyarerisk-
completeinformationbytwoplayerswhosepayofffunctionsare neutral;thusðJ ,J Þaretakenasdollarvalues.).Itisalsopossible
1 2
commonknowledge;thatis,knowntobothofthem.Forthisclass torepresentthisgamewithdiscretestrategiesusingagametree
ofsimplegameswefirstconsiderstaticgameswheretheplayers asinFig.1.Notethatboth(a)and(b)inthisfigureareequivalent
choose their strategies simultaneously. We then consider a representations of the simultaneous game where the nodes
dynamic (two-stage) game, where the players choose their connected by a dashed line constitute a player’s information set.
strategies sequentially. For the static case, the solution concept In Fig. 1(a), when P2 makes a move, she is at a node in the
is the Nash equilibrium which we compute using the best information set indicated, but she does not know which node
response analysis. For the dynamic case, the solution concept is sinceinthesimultaneousgameP1wouldnotrevealhischoiceto
subgame perfect equilibrium (SPE) which is computed using P2. Fig. 1(b) has essentially the same interpretation where the
backward induction. We illustrate each solution concept by nodes connected by the dashed line constitute P1’s information
discussing two examples; one with discrete strategies and set.Gametreesplayacrucialroleinidentifyingtheequilibrium
anotherwithcontinuousstrategies. indynamicgames,buttheyarelessusefulinstaticgameswhich
arefrequentlyanalyzedusingthenormalform.
2.1. ModelI:staticgameswithcompleteinformation Instaticgamesofcompleteinformationwhicharerepresented
in normal form, each player has exactly the same number of
Considertwocompetingnewsvendors(denotedbyP1andP2) actions(i.e.,moves)asthenumberofstrategies.Inthisexample,
who face random demand for their product. The newsvendors bothP1andP2havetwoactions/strategiestochoosefrom,thus
(alsocalled‘‘players’’)maylosecustomerstoeachotheriftheydo thenormalformTable1simplyconsistsoftworows(forP1)and
notordersufficientstock.Thus,P1’sexpectedprofitJ1isaffected two columns (for P2). When games become sequential and/or
they involve incomplete information, one needs to distinguish
Table1 between ‘‘actions’’ and ‘‘strategies’’. We will have more to say
Payofftableforthetwonewsvendors’expectedprofitsfor aboutthisdistinctioninsubsequentsections.
the static game where the players make their moves
How should the newsvendors choose their order quantities
simultaneously.Here,eachplayerhastwostrategies.
(i.e.,determinetheirmoves)recognizingthateachnewsvendor’s
P1/P2 ‘2 h2 expected profit depends on both players’ decisions? To answer
this question we determine the best response of each player to
L1 (3,1) - (6,2n) theother’sdecision.IfP1choosesL ,thenP2shouldchooseh as
k k 1 2
H1 ð5n,4nÞ ’ (7n,3) t1hiifsschheohicaedgcivheossehner‘a.n(Wexepiencdteicdatperothfiitsowfi2thwahnicahstiesrhisikghneprlathcaend
2
Fig.1. TwoequivalentgametreerepresentationsofthestaticgamebetweentwoplayersP1andP2.Thenodesconnectedbyadashedlinerepresenttheinformationsetof
aplayer.
H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577 565
nextto2.)Similarly,P2’sbestresponsetoH is‘ indicatedby4n. unsatisfied customers may switch to the other newsvendor if
1 2
WhatareP1’sbestresponsestoP2’smoves?IfP2chooses‘ ,itis he/shehasanyunitsavailable.Forsimplicityofexpositioninthis
2
bestforP1 tochoose H and receive 5n, andif h is chosen,it is paper, we assume thatboth the salvage value and the penalty
1 2
stillbestforP1tochooseH andreceive7n.TheH ‘ cellinthe costs are zero. With these assumptions, the expected profit
1 1 2
secondrowandfirstcolumnofthetableissignificantasthisgives functionofthefirstnewsvendor(P1)isgiveninParlar(1988)as
theNashequilibriumforthisproblem.Thedirectionsofthearrows Z q1 Z 1
inTable1indicatethatanymovementawayfromtheequilibrium J1ðq1,q2Þ¼s1 xfðxÞdxþs1q1 q1fðxÞdx
will not last long and the players will eventually settle at the 0 q1
equilibriumsolutionofH1‘2 withpayoffsðJ1,J2Þ¼ð5,4Þ. þs1Z q1Z Bbðy(cid:3)q2ÞhðyÞfðxÞdydx
TheNashequilibriumisan‘‘equilibrium’’inthesensethatthe 0 q2
players would have no incentive to deviate away from it. For Z q1Z 1
þs ðq (cid:3)xÞhðyÞfðxÞdydx(cid:3)c q , ð3Þ
example,ifP1movestoL ,thenP2wouldplayh resultingina 1 1 1 1
1 2 0 B
reduction of P1’s payoff to 3. But if P2 plays h , then P1 would
2 wheres istheunitsalesrevenue,c istheunitpurchasecostand
preferH inwhichcaseP2wouldchoose‘ thusendingupinthe 1 1
1 2 B(cid:2)ðq (cid:3)xÞ=bþq ,withbasthefractionofP2’sdemandthatwill
‘‘equilibrium’’again. 1 2
switch to P1’s product when P2 is sold out. The second news-
Weformalizetheabovediscussionwiththefollowingdefini-
vendor’sexpectedprofitisobtainedsimilarlyas
tion:theNashequilibriumforatwo-playernon-cooperativegame
isapairðqN,qNÞwiththepropertythat Z q2 Z 1
1 2 J2ðq1,q2Þ¼s2 yhðyÞdyþs2q2 q2hðyÞdy
J ðqN,qNÞZJ ðq ,qNÞ forallq , ð1Þ 0 q2
1 1 2 1 1 2 1 Z q2Z A
þs aðx(cid:3)q ÞfðxÞhðyÞdxdy
2 1
J2ðqN1,qN2ÞZJ1ðqN1,q2Þ forallq2: ð2Þ 0 q1
Z q2Z 1
þs ðq (cid:3)yÞfðxÞhðyÞdxdy(cid:3)c q , ð4Þ
2 2 2 2
0 A
Remark 1. Computingtheequilibriainthe extensiveform:Wecan
wheres andc aretheunitsalesrevenueandunitpurchasecost,
2 2
alsoquicklyidentifytheNashequilibriumbydeterminingwhether
respectively, and A(cid:2)ðq (cid:3)xÞ=aþq , with a as the fraction of P1’s
2 1
a givenstrategyispartofanequilibriumwithoutfirstcomputing
demandthatwillswitchtoP2’sproductwhenP1issoldout.
thenormalformofthegame.Thismethodbecomesimportantin
Parlar(1988)hasshownthat
identifying the equilibria in games of incomplete information. To
iwlleusstereattehtahtisPlmayeetrhoP2d,’sabsseustmreestphoantsPe1tochtohoissecshoL1ic.eFriosmh2Fbige.ca1u(as)e, @@qJ11 (cid:2)I1ðq1,q2Þ¼s1Zq11fðxÞdxþq1Z0q1fðxÞdx
2isbetterthan1.ButifP1realizesthatP2willchooseh ,weagain Z q1Z 1
2 þs hðyÞfðxÞdydx(cid:3)c , ð5Þ
seefromFig.1(a)thatnowP1would playH because7isbetter 1 1
1 0 B
than6;thusthegivenstrategyL cannotbepartofanequilibrium.
NowassumethatP1choosesH11towhichP2’sbestresponsewould @J2 (cid:2)I ðq ,q Þ¼s Z 1hðyÞdyþq Z q2hðyÞdy
be ‘2 because 4 is better than 3. But then P1 would have no @q2 2 1 2 2 q2 2 0
incentivetomove away from H1 because 5isbetter than3,thus þs Z q2Z 1fðxÞhðyÞdxdy(cid:3)c , ð6Þ
resultingintheequilibriumH ‘ withthepayoffs(5,4). & 2 2
1 2 0 A
We note that even though we have considered (and found) andthat@I1=@q1¼@2J1=@q21o0,indicatingthestrictconcavityofJ1
only thepure strategies forthe game represented in Table 1, for foreachq .(Usingparallelarguments,itisalsopossibletoshow
2
somediscretegamesapurestrategymaynotexist.Forsuchcases, that@2J2=@q22o0,i.e.,J2isstrictlyconcaveforeachq1.)Employing
wecandeterminemixedstrategieswhoseexistencewasprovenin these results, Parlar (1988) proves the uniqueness of the Nash
Nash(1950)foranygamewithfinitenumberofpurestrategies. equilibriumforthisproblem.
To illustrate the above results, let us assume that demand
2.1.2. Continuousstrategies densities are exponential, i.e., fðxÞ¼le(cid:3)lx and hðyÞ¼me(cid:3)my with
WhentheexpectedprofitfunctionsJiðq1,q2Þarecontinuousin respective parameters ½l,m(cid:4)¼½1 , 1(cid:4), and means EðXÞ¼30 and
30 20
the strategies, the best response functions and the Nash equili-
EðYÞ¼20. The other parameters are given as ½a,bjs ,s jc ,c (cid:4)¼
1 2 1 2
briumaredeterminedasfollows:GivenP2’sstrategyq ,P1must
2 ½0:9,0:9j15,9j8,5(cid:4). With these data values, we find the news-
maximizehisexpectedprofit;thusP1findshisbestresponseby
vendors’expectedprofitsas
maximizing J ðq ,q Þ for a given q . That is, P1 must solve
1 1 2 2
maxq1 J1ðq1,q2Þfor eachpossiblevalueof q2 and obtain R1ðq2Þ as J1ðq1,q2Þ¼450ð1(cid:3)e(cid:3)q1=30Þþ270e(cid:3)q2=20þ405e(cid:3)ðq1=18þq2=20Þ
his best response. Similarly, P2 solves maxq2 J2ðq1,q2Þ for each (cid:3)675e(cid:3)ðq1=30þq2=20Þ(cid:3)8q1,
possible value of q and obtain R ðq Þ as her best response.
1 2 1
Provided that the payoff functions Jiðq1,q2Þ are continuously J2ðq1,q2Þ¼180ð1(cid:3)e(cid:3)q2=20Þþ243e(cid:3)q1=30(cid:3)65761e(cid:3)ðq1=30þq2=27Þ
differentiableintheirargumentqiandconcaveforeveryqj,iaj, þ48760e(cid:3)ðq1=30þq2=20Þ(cid:3)5q2:
i,j¼1,2, the best response function is found from
@Ji=@qi(cid:2)Iiðq1,q2Þ¼0. It then follows that a Nash equilibrium (if NotethatifP1couldchoosebothq1andq2atwill,hewould
itexists)isfoundasasolutionofthesystemoftwoequations solve the optimization problem maxq1,q2 J1ðq1,q2Þ and obtain
ðq ,q Þ¼ð37:21,0Þ and J ¼148:11. Naturally, in a competitive
@J @J 1 2 1
I ðq ,q Þ(cid:2) 1 ¼0 and I ðq ,q Þ(cid:2) 2 ¼0: setting P1 has no control over P2’s order quantity and thus this
1 1 2 @q 2 1 2 @q
1 2 solutionwouldnotbepossible.Similarly,ifP2couldchooseP1’s
Toillustratetheseresults,considerasimplifiedversionofthe orderquantity,thenshewouldsolvemax J ðq ,q Þandobtain
q1,q2 2 1 2
competitivenewsvendormodeldiscussed inParlar(1988).Asin ðq ,q Þ¼ð0,35:21ÞandJ ¼80:98,butthissolutionwouldalsonot
1 2 2
thediscretestrategyexamplediscussed above,thenewsvendors bepossiblesinceP2hasnocontroloverP1’sdecisions.
facerandomdemandsXandYwithrespectivedensitiesf(x)and TodeterminetheNashequilibriumforthisproblemwhereP1
h(y) and if one newsvendor runs out of stock, some of the chooseshisstrategyq andP2choosesherstrategyq weproceed
1 2
566 H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577
asfollows:Differentiatingtheexpectedprofitfunctionswehave the leader’s moves and thus the follower must have a complete
planofactionspecifiedforallthepossibilitiesthatshemayface.
@J 45 45
@q1 (cid:2)I1ðq1,q2Þ¼15e(cid:3)q1=30(cid:3) 2 e(cid:3)ðq1=18þq2=20Þþ 2 e(cid:3)ðq1=30þq2=20Þ(cid:3)8, This means that P2 now has a total of four strategies available
1
giveninTable2.
ð7Þ
For ease of reference, we use the ‘‘mapsto’’ notation / to
@J 243 243 denote P2’s strategies as a function of P1’s moves. For example,
@q22 (cid:2)I2ðq1,q2Þ¼9e(cid:3)q2=20þ 7 e(cid:3)ðq1=30þq2=27Þ(cid:3) 7 e(cid:3)ðq1=30þq2=20Þ(cid:3)5: P2’sfirststrategyisdenotedbyðL1,H1Þ/‘2‘2whichindicatesthat
ð8Þ P2 will always choose ‘2 regardless of what P1 does; and P2’s
thirdstrategyisdenotedbyðL ,H Þ/h ‘ whichindicatesthatP2
1 1 2 2
ThebestresponsefunctionR ðq ÞforP1canbeextracted(inthis
1 2 will choose h if P1 chooses L , and she will choose ‘ if P1
2 1 2
case, numerically) by solving I ðq ,q Þ¼0 for each value of q .
1 1 2 2 choosesH .Thus,astrategyforaplayerisacompleteplantoplay
1
Similarly, the best response function R ðq Þ for P2 can be
2 1 thegame.(NotethatifP1hadthreemovesandP2hadtwo,then
extracted (again, numerically) by solving I2ðq1,q2Þ¼0 for each P2 would have a total of 23¼8 strategies. In general, with m
valueofq .Thus,tocomputetheequilibriumwesolvethesystem
1 movesforP1andnmovesforP2,theformerhasmstrategiesand
of two equations I1ðq1;q2Þ¼0 and I2ðq1;q2Þ¼0 for the two the latter has nm strategies. See, Peters (2008, Chapter 4.2) and
unknowns.ThisgivesðqN,qNÞ¼ð25:38,19:55ÞastheuniqueNash
1 2 Webb(2007,Chapter2.2)foragooddiscussionoftheenumera-
equilibrium with J ðqN,qNÞ¼83:63 and J ðqN,qNÞ¼35:91. In this
1 1 2 2 1 2 tionofstrategiesingameswithfinitenumberofmovesforeach
competitivescenarioeachplayerreceivesalowerexpectedprofit
player.)
than what they would have obtained if they could have chosen
Given the two strategies (moves) available to P1 and four
bothdecisionvariablesfreely.
strategiesavailabletoP2, wecanestablishthenormalformalof
this dynamic game and attempt to identify the equilibrium. The
2.2. ModelII:dynamicgameswithcompleteinformation normal form is given in Table 3 where the first and the last
columnsareexactlythesameasthefirstandsecondcolumnsof
Wenowconsidera dynamic(two-stage)versionof thegame thenormal form matrix for the simultaneous game discussed in
discussedabove.WhereasinSection2.1theplayerswerechoos- Section 2.1 (Table 1). Note, for example, that the strategy
ing their strategies simultaneously, now the decisions are made combination ðL ,‘ ‘ Þ results in payoff vector (3,1) since P1’s
1 2 2
sequentially(perhapsbecauseoneoftheplayerscanactquickly choiceofL isfollowedbyP2’schoiceof‘ .Similarly,thestrategy
1 2
and make his decision before the other one). Retaining the combination ðH ,h ‘ Þ results in payoff vector (5,4) since P1’s
1 2 2
assumption of complete information, we now examine the choiceofH isfollowedbyP2’schoiceof‘ .
1 2
resulting complications arising from the sequential nature of It is easy to see that the direction of arrows in Table 3 show
thegame. thatthisgamehastwoNashequilibria;onebeingðH ,‘ ‘ Þwith
1 2 2
payoffsð5n,4nÞ;andtheotherðL ,h ‘ Þwithpayoffsð6n,2nÞ.Which
1 2 2
2.2.1. Discretestrategies equilibrium is the one that will/should be used in this dynamic
Let us return to the same problem we discussed in Section game?Beforeweanswerthisquestion,wepointoutthatwecan
2.1.1butnowsupposethatP1actsbeforeP2inchoosinghisorder determinewhichstrategywillbepartofanequilibriumwithout
quantitystrategy;thusinthisversionofthegameP1becomesthe computingthenormalform.
‘‘leader’’ and P2 becomes the ‘‘follower.’’ Will this result in a
‘‘first-moveradvantage’’forP1?Toanswerthisquestion,wefirst
Remark 2. Computing the equilibria in the extensive form: We
consider the extensive form representation of the game given in
observe here that one could again quickly determine whether a
Fig.2whereP1movesfirstandP2movesnext.Dependingonthe
strategyispartofaNashequilibriumwithoutfirstcomputingthe
combinationofmovesmadebytheplayers,thepayoffsobtained
normalformofthegame—aswasdoneinSection2.1.1.Whenwe
areindicatedattheendpointsofthegametree.Indynamicgames
consider L we note that P2’s best responses to L are both h ‘
withsequentialdecisions,asubgameisdefinedasthatpartofthe 1 1 2 2
andh h .Facedwithh ‘ ,P1willhavenoincentivetodeviateto
gametreethatstartsataparticularnodeoftheoriginalgame.For 2 2 2 2
H (sincethatwouldresultinreceiving5ratherthan6),butfaced
example,oneofthesubgamesinFig.2startsatnodeindicatedby 1
/2aS;andanothersubgamestartsat/2bS.Thecompletegame
itselfwhichstartsatnode/1Sisalsoconsidereda‘‘subgame.’’ Table2
StrategiesforplayerP2asafunctionofplayerP1’smoves.
Now that the structure of the game has changed and the
choicesaremadesequentially,P2nolongerhas‘2 andh2asthe P1’smoves P2’sstrategies
onlystrategies(whichwasthecaseinthestaticgameofSection
2.1). In dynamic games the follower’s moves are conditional on #1 #2 #3 #4
L1 ‘2 ‘2 h2 h2
H1 ‘2 h2 ‘2 h2
Notation ðL1,H1Þ/‘2‘2 ðL1,H1Þ/‘2h2 ðL1,H1Þ/h2‘2 ðL1,H1Þ/h2h2
Table3
Payofftableforthetwonewsvendors’expectedprofitsforthedynamicgame.In
thistableP1isthe‘‘leader’’andP2isthe‘‘follower’’wheretheformerhastwo
strategies,butthelatterhasfourstrategies.
P1/ ðL1,H1Þ/‘2‘2 ðL1,H1Þ/‘2h2 ðL1,H1Þ/h2‘2 ðL1,H1Þ/h2h2
P2
L1 (3,1) 2 (3,1) - ð6n,2nÞ 2 ð6,2nÞ
k k m k
Fig.2. Extensiveformofthedynamicgamewithtwoplayerswherethesubgame H1 ð5n,4nÞ ’ ð7n,3Þ - ð5,4nÞ ’ ð7n,3Þ
perfectequilibriumisfoundasðL1;h2‘2Þ.
H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577 567
with h h , P1 will deviate to H . Thus, ðL ,h ‘ Þ must be an
2 2 1 1 2 2
equilibrium.Similarly,ifP1choosesH ,thenP2’sbestresponses
1
are‘ ‘ andh ‘ ,butP1willnotdeviatefromH whenfaced‘ ‘ ,
2 2 2 2 1 2 2
but will deviate when faced with h ‘ . Thus, ðH ,‘ ‘ Þ is also a
2 2 1 2 2
Nashequilibrium. &
Returningtothequestionofwhichequilibriumistheonethat
will/shouldbeusedinthisdynamicgame,werefertoFig.2and
observethatifthegameeverarrivedatnode/2aS,itisoptimal
forP2tochooseh withapayoffofð6,2nÞ,whichisbetterforher
2
thanð3,1Þ;andifitarrivedat/2bSitisoptimalforP2tochoose
‘ with a payoff of ð5,4nÞ, which is better than (7,3). That is, the
2
bestresponseR ðq ÞforP2isgivenas
2 1
(
h if q ¼L ,
2 1 1
R ðq Þ¼ ð9Þ
2 1 ‘ if q ¼H :
2 1 1
Moving back to node /1S, P1’s problem is now to maximize
J ðq ,R ðq ÞÞwhereP2’sbestresponseR ðq Þisgivenin(9).Thus,
1 1 2 1 2 1
P1 makes his choice by comparing (6,2) (if he chooses L ) and
1
(5,4)(ifhechoosesH ),resultingintheoptimalchoiceofL .This
1 1
method of solving the game is known as rollback (Dixit et al.,
2009,Chapter3)orbackwardinduction(Gibbons,1992,Chapter2) Fig.3. TheStackelbergequilibriumisobtainedbysolvingthefollowingproblem:
which uses the same principle as dynamic programming maxJ1ðq1,q2ÞsubjecttoI2ðq1,q2Þ¼0.ThisfiguresuperimposesthecontoursofP1’s
(Bellman, 1957). The backward induction equilibrium is shown objectiveJ1ðq1,q2ÞwithP2’sbestresponseI2ðq1,q2Þ¼0.Intheequilibriumwehave
ðqS,qSÞ¼ð28:38,18:60Þ.
asthicklinesinFig.2andisdenotedbyðL ,h ‘ Þsinceitisoptimal 1 2
1 2 2
forP1tochooseL at/1S,anditisoptimalforP2tochooseh if
1 2
P1choosesL ,and‘ ifP1(ever)choosesH . Thus,P1mustchoosehisorderquantitythatmaximizesJ ðq ,q Þ
1 2 1 1 1 2
Note that at each subgame the backward induction principle subject to the constraint I ðq ,q Þ¼0. For this game, the Stackel-
2 1 2
producesachoicethatisoptimalforthatplayerresultinginthe berg solution for P1 is obtained by solving the optimization
equilibrium for the game. In games with finite trees and perfect problem
information (where the players know the result of all previous
maxJ ðq ,q ÞsubjecttoI ðq ,q Þ¼0,
moves, as the ones considered in this paper) the equilibrium q1Z0 1 1 2 2 1 2
foundbythebackwardinductionprincipleisalsocalledsubgame-
where J ðq ,q Þ and I ðq ,q Þ¼0 are given in (3) and (8), respec-
perfectequilibrium(SPE).Moreformally,asubgame-perfectequili- 1 1 2 2 1 2
tively.Thisproblemisshownintwodimensionsbyprojectingthe
brium is defined as a combination of strategies for both players
contours of J ðq ,q Þ onto the ðq ,q Þ plane and by choosing the
thatresultinaNashequilibriumineverysubgamewhichspecify 1 1 2 1 2
highestvaluedcontourthatistangenttoP2’sbestresponsecurve
movesthat are bestresponses to each other(as we sawabove);
I ðq ,q Þ¼0asinFig.3.Solvingtheoptimizationproblemwiththe
see,Dixitetal.(2009,Chapter3),Gibbons(1992,Chapter2),and 2 1 2
same data as in Section 2.1.2 gives the Stackelberg solution as
Selten(1965)(wheretheconceptwasfirstintroduced).
ðqS,qSÞ¼ð28:38,18:60Þ and ðJ ðqS,qSÞ,J ðqS,qSÞÞ¼ð84:35,33:94Þ.
Insummary,thesubgame-perfectequilibriuminourexample 1 2 1 1 2 2 1 2
isðL ,h ‘ Þwhichcanalsobewrittenasðqn,R ðq ÞÞwhereqnisP1’s ThisresultindicatesthatP1hasobtainedafirst-moveradvantage
opti1ma2l 2choice at node /1S. It is impo1rtan2t 1to note tha1t if P1 compared to the Nash solution since J1ðqS1,qS2Þ4J1ðqN1,qN2Þ, which
chooses his optimal strategy L , then P2’s optimal move is to has resulted in a second-mover disadvantage for P2 since
1
chooseh2,andthustheequilibriumpatharisingfromSPEisL1-h2. J2ðqS1,qS2ÞoJ2ðqN1,qN2Þ.
Thus,eventhough‘ isapartoftheSPE,P2willneverchooseit
2
unlessP1makesanonoptimaldecisionatnode/1Sandchooses
H at that node. In summary, the SPE ðL ,h ‘ Þ is the backward 3. ModelIII:staticgameswithincompleteinformation
1 1 2 2
induction equilibrium, whereas ðL ,h Þ is the backward induction
1 2
outcome that is on the equilibrium path. If both players choose ThegamesofcompleteinformationdescribedinModelsIand
theirstrategiesoptimally,thentheresultingsequenceofdecisions IIabovehadthecommonfeaturethatbothplayerswereinformed
L -h that are on the equilibrium path is known as Stackelberg about each other’s payoff functions. In a game of incomplete
1 2
solution (vonStackelberg, 1934) which we denote by ðqS,R ðqSÞÞ. information players may not know the payoff function of some
1 2 1
Thus, the (rational) choice of the players leading to the equili- otherplayer,ortheymaynotknowwhatactionsareavailableto
briumpatheliminatestheotherNashequilibrium,i.e.,ðH ,‘ ‘ Þ. other player(s). For example, even though P2 would know her
1 2 2
ownpurchasecostc withcertainty,shemayonlyknowthatP1’s
2
purchasecost is c (low)with probabilityy , or c (high) with
2.2.2. Continuousstrategies 1L 1 1H
probability1(cid:3)y .SinceP1knowshiscost(whichiseitherc or
Letusnowreturntothenewsvendorproblemwithcontinuous 1 1L
c ), he has superior information, i.e., information structure is
strategiesdiscussedinSection2.1.2andassumethatP1isthefirst 1H
asymmetricinfavorofP1.
mover(the‘‘leader’’)andP2isthesecondmover(the‘‘follower’’)
Using Harsanyi’s (1967) approach elucidated in a three-part
withrespectiveobjectivefunctionsJ ðq ,q ÞandJ ðq ,q Þgivenin
1 1 2 2 1 2
essay, we solve the resulting Bayesian game by assuming that
(3)and(4).Forthisgame,thebestresponsefunctionR ðq ÞforP2
2 1
every player can be of several possible types, where a type
is obtained by maximizing J ðq ,q Þ for each q , or equivalently,
2 1 2 1
summarizes all relevant information about a player such as the
solving@J =@q ¼I ðq ,q Þ¼0foreachvalueofq ;thatis,
2 2 2 1 2 1
payoffs and possible moves. (See Myerson (2004) for an illumi-
R ðq Þ¼argmaxJ ðq ,q Þ¼(cid:2)q : @J2 ¼I ðq ,q Þ¼0(cid:3): natingcommentaryonHarsanyi’sthreepapers.)Insuchagame,
2 1 q2Z0 2 1 2 2 @q2 2 1 2 Harsanyi’s (1967) transformation introduces an artificial player,
568 H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577
called Nature, which chooses a particular type for all players Whentheplayersadoptthestrategyprofileðs ðt Þ,s ðt ÞÞwe
1 1 2 2
(according to some joint probability distribution) and reveals to definetheconditionalexpectedpayoffsforplayerPioftypet as
i
playerPihistypeti.Thus,someplayerscannotobservethemove J^ ðs ðt Þ,s ðt Þ,t Þ¼ XJ ðs ðt Þ,s ðt Þ;t ,t Þp ðt jt Þ
ofNatureregardingtheactualtypeoftheotherplayer(s),butthey 1 1 1 2 2 1 1 1 1 2 2 1 2 1 2 1
knowthejointprobabilitydistributionfromwhichNatureselects t2AT2
forallt AT , ð10Þ
thetypes.Thismeansthattheplayersfaceagamewithimperfect 1 1
information (because of the uncertainty about the move of the J^ ðs ðt Þ,s ðt Þ,t Þ¼ XJ ðs ðt Þ,s ðt Þ;t ,t Þp ðt jt Þ
‘‘player’’ Nature), and hence the incompleteness of information 2 1 1 2 2 2 2 1 1 2 2 1 2 2 1 2
aboutpayoffsistransformedintouncertaintyaboutthemoveof t1AT1
forallt AT , ð11Þ
Nature. 2 2
Forexample,ifP1’spurchasecostiseitherc1Lorc1H,thenthis whereJiðs1ðt1Þ,s2ðt2Þ;t1,t2ÞisplayerPi’spayoffwhenthisplayer’s
playerwillhavetwotypesT1¼fc1L,c1Hg,andifP2’spurchasecost typetiadaptsthestrategysiðtiÞfori¼1,2.Withthisdefinition,a
isc2,thenP2willhaveonetypeT2¼fc2g,only.Itisassumedthat strategyprofileðsn1ðt1Þ,sn2ðt2ÞÞisa(pure)BayesianNashequilibrium
each player knows his/her type, and given this, each player can ofastaticBayesiangameifforeachplayerPi,everytypetAT of
i i
computehis/herbeliefsonthetypesofotherplayers.Thatis,for playerPi,andeveryalternativestrategys0ðtÞofplayerPi,wehave
i i
t AT and t AT , the beliefs p ðt jt Þ and p ðt jt Þ are computed
1 1 2 2 1 2 1 2 1 2 J^ ðsnðt Þ,snðt Þ,t ÞZJ^ ðs0ðt Þ,snðt Þ;t Þ, ð12Þ
bytheconditionalprobabilityformula(i.e.,Bayes’rule)as, 1 1 1 2 2 1 1 1 1 2 2 1
p ðt jt Þ¼pðt2,t1Þ and p ðt jt Þ¼pðt1,t2Þ, J^2ðsn1ðt1Þ,sn2ðt2Þ,t2ÞZJ^2ðsn1ðt1Þ,s02ðt2Þ;t2Þ: ð13Þ
1 2 1 pðt Þ 2 1 2 pðt Þ
1 2 SimilartothedefinitionoftheNashequilibriumgivenin(1)and
where the joint probability pðt ,t Þ is assumed common (2), this definition states that whatever a player’s type is, this
1 2
knowledge. player’sstrategyisa bestresponseto thestrategies oftheother
Harsanyi’s (1967) approach assumes that the strategy for player.
playerPiisafunctionsðtÞforeachtypetAT whichspecifiesa
i i i i
feasibleaction.Forexample,ifP1hastwotypes,andheonlyhas 3.1. Discretestrategies
twomovesfL ,H g,thenhehasfourpossiblestrategies.Theseare
1 1
listedinTable4whereL L indicatesthatP1willchooseL when To illustrate the above discussion of the Bayesian Nash
1 1 1
his type is either c or c (that is, he always chooses L ); L H equilibrium, let us consider a discrete strategy problem where
1L 1H 1 1 1
indicatesthatP1willchooseL ifhistypeisc ,willchooseH if eachplayerhastwopossiblemoves(i.e.,P1orderslowL orhigh
1 1L 1 1
histypeisc ,etc. H ,andP2orderslow‘ orhighh ).ButnowP1knowsthathis
1H 1 2 2
Harsanyi(1967)proposedmodelingstaticgameswithincom- type(purchasecost)iseitherc (low)orc (high),butP2only
1L 1H
pleteinformationbyincludingNatureasanimaginaryplayer.In knows the probability distribution of P1’s type, that is, that
the above game Nature moves first and determines P1’s type Prð P1’s type is c Þ¼y and Prð P1’s type is c Þ¼1(cid:3)y . P2 has
1L 1 1H 1
whichisc withprobabilityy andc withprobability1(cid:3)y .P1 onlyonetype,c ,andbothplayersknowthis.
1L 1 1H 1 2
knows his type (i.e., his payoffs), but P2 knows only that she is The game trees for this game played under incomplete
facinganopponent(P1)whosepurchasecostiseitherc orc information are represented in Fig. 4. Player P1 knows exactly
1L 1H
withprobabilitiesy and1(cid:3)y ,respectively.Oneinterpretationof whichgameisbeingplayed,butP2knowsthatthegameonthe
1 1
this game is that P1 and P2 are randomly paired and the leftwillbeplayedwithprobabilityy andthegameontheright
1
proportionoflowcostP1sisy .SinceP2hasonlyonetype,the withprobability1(cid:3)y .Onecaninterpretthisgameofincomplete
1 1
conditional probabilities for P1 are simply p ðt jc Þ¼1 and information by saying that as far as P2 is concerned P1 has two
1 2 1L
p ðt jc Þ¼1. However, since P1 has two types, P2’s conditional ‘‘personalities’’andP2facesthetype1personality(purchasecost
1 2 1H
probabilitiesarep ðc jt Þ¼p ðc Þ¼y ,andp ðc jt Þ¼p ðc Þ¼ c )withprobabilityy andtype2personality(purchasecostc )
2 1L 2 2 1L 1 2 1H 2 2 1H 1L 1 1H
1(cid:3)y .ItisimportanttonotethateventhoughP1knowshistype, withprobability1(cid:3)y .
1 1
the Bayesian equilibrium solution must still provide a complete TheHarsanyitransformationinvolvesconvertingthegameof
planofactionforbothplayers,i.e.,inthediscreteversionofthe incomplete information to a game of complete but imperfect
game,P1mustconsiderhisfourstrategiesfL L ,L H ,H L ,H H g information. The game tree for the transformed version of the
1 1 1 1 1 1 1 1
shown in Table 4, and P2 must consider her two strategies problem is given in Fig. 5 where Nature moves first and deter-
(moves)f‘ ,h g. minesP1’stype,i.e.,loworhighpurchasecost.(Wewillusethe
2 2
Fig.4. TheincompleteinformationgamewhereP1knowswhichgameisbeingplayed,i.e.,eithertheoneontheleftwherehistypeisc1L,ortheoneontherightwherehis
typeisc1H.P2onlyknowstheprobabilitydistributionofP1’stype,i.e.,ðy1,1(cid:3)y1Þ.
H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577 569
Fig.5. TheincompleteinformationgameinFig.4becomesanequivalentcompletebutimperfectinformationgamethroughtheHarsanyitransformation.
Table4 TheotherexpectedconditionalpayoffsinTable5arecalculatedin
StrategiesforplayerP1asafunctionofhistype. asimilarmanner.
TodeterminetheBayesianNashequilibriumforthisgame,we
P1’sstrategies P1’stypet1 usethefamiliarapproachand identify thebestresponses forP2
c1L c1H Notation foreachpossiblestrategyofP1.Forexample,ifP1usesH1L1,itis
bestforP2touse‘ sincethisgivesheranexpectedpayoffof3n
2
##12 LL11 LH11 ððcc11LL,,cc11HHÞÞ//LL11LH11 f(orautnhderbtyhainde2n.5tifiyfinhg2 itshuesesdtr)a.tSeigmyilathrlayt, Pg1iv’sesbehsitmretshpeonhseigshaerset
##34 HH11 LH11 ððcc11LL,,cc11HHÞÞ//HH11LH11 expected payoff, given P2’s strategy. For example, if P2 chooses
‘ ,thenthebestresponseforP1iseitherL H orH H .Thus,the
2 1 1 1 1
pure strategy equilibria in this game are found as ðL H ,‘ Þ and
1 1 2
same payoff values we have in Fig. 5 when we consider the ðH H ,‘ Þwithpayoffsof(4,3)toP1and2toP2inthefirstcase;
1 1 2
dynamicversionof this problem withincomplete informationin andð4,3ÞforP1and3forP2inthesecondcase.Inthisproblem
Section4.)OnceNaturemakesherchoice,P1knowshistypebut withmultiple equilibria, ðL H ,‘ Þ may bethe one that is imple-
1 1 2
thisinformationisnotrevealedtoP2.WiththeinclusionofNature mented if P1 wants to see P2 receive the least amount. On the
as one of the players, the game becomes one of complete otherhand,ifnegotiationsarepossibleandifP2canmotivateP1
informationsince theplayers know all the payoffs ontheexten- tomovetoH H withthepotentialpromiseofaside-paymentto
1 1
sive form; the game also becomes one of imperfect information P1, then ðH H ,‘ Þ could be the equilibrium of the game. (See
1 1 2
becauseP2willnotbeawareofwhatNaturehaschoseninitially. Schelling, 1960, pp. 54–58, 89–118 for an explanation of the
We now represent this extensive game of complete but conceptofafocalpointingameswithmultipleNashequilibria.)
imperfect information to an equivalent strategic form game and The above procedure illustrates Harsanyi’s insight (1967)
determine the Bayesian Nash equilibrium. First, recall that, as whichtransformsanincompleteinformationgame(asshownin
indicated in Table4,P1 has a total of four strategies as he must Fig.4)toacompletebutimperfectinformationgame(asshownin
haveacompleteplanofactionfollowingNature’schoice.Onthe Fig.5).WhilethegameinFig.4involvesP2’suncertaintiesabout
otherhand,sinceP2hasonetypeonly(purchasecostofc ),this P1’spurchasecost—andhencetheincompletenessofinformation—
2
playerstillhastwostrategies,oneforeachpossiblemove‘ and thestrategicformgameofTable5doesnotinvolveanyuncertainty
2
h . The (expected) payoffs for all possible combinations of all because it is subsumed in the expected payoff calculations for P2.
2
strategy pairs are given in Table 5. Consider, for example, the However,thegameinFig.5isofimperfectinformationvarietysince
strategy combination ðL L ,h Þ, that is, regardless of Nature’s P1andP2makesimultaneousdecisionsafterNaturemovesandP2
1 1 2
choice, P1 always orders low ðL L Þ, and P2 orders high ðh Þ. doesnotknowwhichmoveNaturehasmadewhenshemakesher
1 1 2
Now, referring to Fig. 5, if P1’s cost is c , then the expected decision.
1L
conditionalpayoffforP1iscomputedas
Remark 3. Computing the equilibria in the extensive form: As in
J^ ðL L ,h ;c Þ¼J ðL L ,h ;c ,c Þp ðc jc Þ¼4(cid:5)1¼4,
1 1 1 2 1L 1 1 1 2 1L 2 1 2 1L previousmodels,itiseasytodeterminewhetheragivenstrategy
sinceP2hasonlyonetype,i.e.,c .Similarly,IfP1’scostisc ,then is part of an equilibrium without computing the payoffs in the
2 1H
theexpectedconditionalpayoffis, normal form of Table 5. First, consider L L for which P2’s best
1 1
responseis‘ .But,ifP2plays‘ ,thenP1’stypec canimprove
J^ ðL L ,h ;c Þ¼J ðL L ,h ;c ,c Þp ðc jc Þ¼4(cid:5)1¼4: 2 2 1H
1 1 1 2 1H 1 1 1 2 1H 2 1 2 1H bydeviatingtoeitherL H orH H ,thusL L cannotbepartofan
1 1 1 1 1 1
Tocompute P2’sexpected payoff, we recall that p2ðc1Ljc2Þ¼ 12 equilibrium.UsingthesamereasoningitcanbeshownthatH1L1
andp2ðc1Hjc2Þ¼ 12whichgives cannot be part of an equilibrium, either. However, if P1 chooses
J^ ðL L ,h ;c Þ¼J ðL L ,h ;c ,c Þp ðc jc Þ L1H1,thenP2’schoiceof‘2doesnotofferanymotivationforP1to
2 1 1 2 2 2 1 1 2 1L 2 2 1L 2 movetoadifferentstrategy,thusL H ispartofanequilibrium.
þJ ðL L ,h ;c ,c Þp ðc jc Þ 1 1
2 1 1 2 1H 2 2 1H 2 SimilarargumentsleadtotheconclusionthatH H mustalsobe
1 1
¼0(cid:5)12þ3(cid:5)12¼1:5: partofanequilibrium.
570 H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577
Table5 @J2=@q2¼I2ð(cid:5),q2Þ, respectively. Thus, the Bayesian Nash equili-
PayofftableobtainedaftertheHarsanyitransformationoftheoriginalpayoffsin briumconditionsof(14)–(16)simplifyto
thediscretestrategy,staticincompleteinformationgame.
I ðq ,q Þ¼0,
1L 1L 2
P1/P2 ‘2 h2
I ðq ,q Þ¼0,
1H 1H 2
ðc1L,c1HÞ/L1L1 ð4n,2Þ,2n ð4n,4Þ,1:5
ðc1L,c1HÞ/L1H1 ð4n,3nÞ,2n ð4n,5nÞ,1:5 y1I2ðq1L,q2Þþð1(cid:3)y1ÞI2ðq1H,q2Þ¼0: ð17Þ
ðc1L,c1HÞ/H1L1 ð4n,2Þ,3n (3,4),2.5
ðc1L,c1HÞ/H1H1 ð4n,3nÞ,3n (3,5n),2.5 Let us now return to the continuous strategy example dis-
cussed in Section 2.1.2 where the demand densities were expo-
nentialwithmeansEðXÞ¼30andEðYÞ¼20.Asbefore,wesetthe
otherparametersas½a,bjs ,s jc (cid:4)¼½0:9,0:9j15,9j5(cid:4),butnowsince
3.2. Continuousstrategies 1 2 2
the unit purchase cost of the first newsvendor could be low or
high, we let ½c ,c (cid:4)¼½6,10(cid:4). With these values the first-order
Let us return to the competitive newsvendor problem dis- 1L 1H
conditionsgivenin(17)reduceto
cussed in Section 2.1.2 but assume now that at least one of the
players is unsure about the actual purchase cost of the other I1Lðq1L,q2Þ¼15e(cid:3)q1L=30(cid:3)425e(cid:3)ðq1L=18þq2=20Þþ425e(cid:3)ðq1L=30þq2=20Þ(cid:3)6,
player.
I1Hðq1H,q2Þ¼15e(cid:3)q1H=30(cid:3)425e(cid:3)ðq1H=18þq2=20Þþ425e(cid:3)ðq1H=30þq2=20Þ(cid:3)10,
3.2.1. PlayerP1hastwotypesandplayerP2hasonetype
Assume here that P2 is not sure about P1’s purchase cost y1I2ðq1L,q2Þþð1(cid:3)y1ÞI2ðq1H,q2Þ
which is either c1L or c1H with probabilities y1 and 1(cid:3)y1. Given ¼9e(cid:3)q2=20þ21443e(cid:3)ðq1L=30þq2=27Þ(cid:3)21443e(cid:3)ðq1L=30þq2=20Þ
thisuncertaintythatP2faces,whatarethebeststrategiesforP1 þ243e(cid:3)ðq1H=30þq2=27Þ(cid:3)243e(cid:3)ðq1H=30þq2=20Þ(cid:3)5:
andP2,i.e.,whatshouldbetheBayesianNashequilibriumorder 14 14
quantitiesforeachnewsvendor? Weplottheimplicitsurfacesforthesefirst-orderconditionsin
Since P1’s cost (types) may be c1L or c1H, this newsvendor’s Fig.6andnotethattheyintersectatauniquepoint.Solvingthis
strategy set is ½0,1Þ(cid:6)½0,1Þ with moves ðq1L,q1HÞ. Similarly, P2 system of three nonlinear equations, we find ½sn1ðt1Þ,sn2ðt2Þ(cid:4)¼
hasonlyonecostc2,thusherstrategysetis½0,1Þwiththemove ½ðq1L,q1HÞ,q2(cid:4)¼½ð35:46,17:02Þ,19:75(cid:4). Comparing this to the result
q2.Now,referringtotheconditionalexpectedpayoffexpressions obtained in Section 2.1.2 where we had found ðqN1,qN2Þ¼
in (10)–(11), we write the first newsvendor’s objective function, ð25:38,19:56ÞastheNashequilibrium,theBayesianNashequili-
conditionalonhiscostas, brium result shows that the first newsvendor should order a
J^ ðs ðt Þ,s ðt ÞÞ¼J ðq ,q Þp ðc jc Þ¼J ðq ,q Þ, higher quantity than before if he has a lower purchase cost
1L 1 1 2 2 1L 1L 2 1 2 1L 1L 1L 2 ðc ¼6Þ,andalowerquantitythanbeforeifhehasahighercost
1L
ðc ¼10Þ. The Bayesian Nash order quantity for the second
J^ ðs ðt Þ,s ðt ÞÞ¼J ðq ,q Þp ðc jc Þ¼J ðq ,q Þ, 1H
1H 1 1 2 2 1H 1H 2 1 2 1H 1H 1H 2 newsvendor is only slightly higher than its Nash counterpart.
where J ðq ,q Þ and J ðq ,q Þ are P1’s expected profits with Substitutingthesevaluesintherespectiveobjectivefunctions,we
1L 1L 2 1H 1H 2
purchase cost c and c , respectively, obtained from (3), and findtheBayesianNashpayoffsas
1L 1H
p ðc jc Þ¼p ðc jc Þ¼1. The second newsvendor’s objective
1 2 1L 1 2 1H ½J^ ðq ,q Þ,J^ ðq ,q Þ(cid:4)¼½143:74,41:23(cid:4)
function must take into account the uncertainty she faces and 1L 1L 2 1H 1H 2
thus,wehave,
J^ ðs ðt Þ,s ðt ÞÞ¼J ðq ,q Þp ðc jc ÞþJ ðq ,q Þp ðc jc Þ
2 1 1 2 2 2 1L 2 2 1L 2 2 1H 2 2 1H 2
¼y J ðq ,q Þþð1(cid:3)y ÞJ ðq ,q Þ,
1 2 1L 2 1 2 1H 2
whereJ ðq ,q ÞandJ ðq ,q ÞareP2’sexpectedprofitswhenP1
2 1L 2 2 1H 2
chooses q and q , respectively, obtained from (4). In our
1L 1H
numericalcalculationswewillsety ¼ 1,asbefore.
1 2
The first newsvendor P1 chooses his Bayesian Nash equili-
brium strategies to maximize J^ ðs ðt Þ,s ðt ÞÞ when his cost is
1L 1 1 2 2
c , and to maximize J^ ðs ðt Þ,s ðt ÞÞ when his cost is c . The
1L 1H 1 1 2 2 1H
othernewsvendorP2choosesq tomaximizeJ^ ðs ðt Þ,s ðt ÞÞ.The
2 2 1 1 2 2
first-order conditions for the equilibrium are determined from
(12)and(13)andarefoundbysolvingasystemofthreenonlinear
equationsinthreeunknownsq ,q andq :
1L 1H 2
@ @
J^ ðs ðt Þ,s ðt ÞÞ¼ J ðq ,q Þ¼0, ð14Þ
@q 1L 1 1 2 2 @q 1L 1L 2
1L 1L
@ @
J^ ðs ðt Þ,s ðt ÞÞ¼ J ðq ,q Þ¼0, ð15Þ
@q 1H 1 1 2 2 @q 1H 1H 2
1H 1H
@ @
J^ ðs ðt Þ,s ðt ÞÞ¼ ½y J ðq ,q Þþð1(cid:3)y ÞJ ðq ,q Þ(cid:4)¼0:
@q 2 1 1 2 2 @q 12 1L 2 1 2 1H 2
2 2
ð16Þ
The explicit expressions for J ðq ,q Þ and J ðq ,q Þ are found Fig. 6. The Bayesian Nash equilibrium solution of Model III with two players
1L 1L 2 1H 1H 2
from (3); the expression for J ð(cid:5),q Þ follows from (4). The first (newsvendors)whereplayer1hastwotypesðc1L,c1HÞ,andplayer2hasonetype
partial derivatives of these fun2ction2s are also available from (5) cðco2rÞr.eTshpeonsdolutotioI1nðqi1sLt,qh2eÞi¼nt0erasnedctiIo1nðq1oHf,tqh2rÞe¼e0s,uarfnadcetshwehheorreiztohnetavlesrtuircfaalcseucrfoarcrees-
and (6) as @J1L=@q1L¼I1Lðq1L,q2Þ, and @J1H=@q1H¼I1Hðq1H,q2Þ, and spondstoy1I2ðq1L,q2Þþð1(cid:3)y1ÞI2ðq1H,q2Þ¼0.
H.Wu,M.Parlar/Int.J.ProductionEconomics133(2011)562–577 571
and partiallydifferentiatingtheexpectedpayoffsandobtain
J^2ðq1L,q1H,q2Þ¼36:30: @q@ J^1Lðs1ðt1Þ,s2ðt2ÞÞ¼y2I1Lðq1L,q2LÞþð1(cid:3)y2ÞI1Lðq1L,q2HÞ¼0,
1L
Thisresultindicatesasubstantialincreaseinexpectedpayofffor
P1 when his cost is low, as should be expected. The second @ J^ ðs ðt Þ,s ðt ÞÞ¼y I ðq ,q Þþð1(cid:3)y ÞI ðq ,q Þ¼0,
newsvendoralsoexpectsaslightlyhigherexpectedpayoff. @q1H 1H 1 1 2 2 21H 1H 2L 2 1H 1H 2H
Wenowpresentabriefdiscussionontheeffectsofuncertainty
@
on the solution, i.e., on the equilibrium order quantitiesand the J^ ðs ðt Þ,s ðt ÞÞ¼y I ðq ,q Þþð1(cid:3)y ÞI ðq ,q Þ¼0,
@q 2L 1 1 2 2 1 2L 1L 2L 1 2L 1H 2L
correspondingexpectedprofits.WhenP2isuncertainaboutP1’s 2L
unitcosts,P2’sexpectedprofitmaydecreaseorincreasedepend- @
ingonwhetherP2ispessimisticoroptimisticaboutP1’scosts,as @q J^2Hðs1ðt1Þ,s2ðt2ÞÞ¼y1I2Hðq1L,q2HÞþð1(cid:3)y1ÞI2Hðq1H,q2HÞ¼0:
2H
we indicate in Table 6. In particular, note that when P2 is
The resulting system of four nonlinear equations in the four
pessimistic, she thinks that P1 is at an advantage because P1’s
costs are ðc ,c Þ¼ð6,8Þwhich is better for P1 than if c ¼8. In unknowns ðq1L,q1H;q2L,q2HÞ can be solved to determine the
1L 1H 1
Bayesian Nash equilibrium for this game where incomplete
thiscaseP2ordersless(17.96vs.19.55)andherexpectedprofitis
information is two-sided. As an example, consider the problem
also lower (32.66 vs. 35.91). On the other hand, when P2 is
optimistic, i.e., when ðc ,c Þ¼ð8,10Þ, P2 orders more and can discussed in Section 3.2.1 with exponential demand densities
1L 1H
having means EðXÞ¼30 and EðYÞ¼20. As before, the other
evenimproveherexpectedprofitto40.11asopposedto35.91.
parameters are ½a,bjs ,s (cid:4)¼½0:9,0:9j15,9(cid:4) and ½c ,c (cid:4)¼½6,10(cid:4),
1 2 1L 1H
but now since the unit purchase cost of the second newsvendor
3.2.2. PlayersP1andP2bothhavetwotypes P2couldbeloworhigh,welet½c ,c (cid:4)¼½3,5(cid:4).Thus,inthiscase
2L 2H
Asafurtherextension,letusnowassumethatnotonlyP2is P2isinabetterpositionthanbeforeashercostcouldbeevenas
uncertainaboutP1’spurchasecost,butalsoP1isuncertainabout low as 3, and hence we would expect P2 to have a higher
P2’s purchase cost. (To our knowledge, this extension has not expectedprofitthanbefore.
been discussed in any of the inventory/supply chain literature.) Solvingtheresultingsystemoffournonlinearequationsgiven
Thatis,asbefore,P1hastwotypesðc1L,c1HÞforwhichP2holdsthe above with ðy1,y2Þ¼ð0:5,0:5Þ, we find the Bayesian Nash equili-
distributionðy1,1(cid:3)y1Þ,butnowP2alsohastwotypesðc2L,c2HÞfor briumas
which P1 holds the distribution ðy ,1(cid:3)y Þ. In other words, the
2 2 ½snðt Þ,snðt Þ(cid:4)¼½ðq ,q Þ,ðq ,q Þ(cid:4)¼½ð33:00,15:35Þ,ð36:80,20:47Þ(cid:4):
conditionalprobabilitiesforP1andP2,respectively,are, 1 1 2 2 1L 1H 2L 2H
The expected profits for each newsvendor, given their type are
p ðc jc Þ¼p ðc jc Þ¼y and p ðc jc Þ¼p ðc jc Þ¼1(cid:3)y ,
1 2L 1L 1 2L 1H 2 1 2H 1L 1 2H 1H 2 computedas
½J ðq ,q Þ,J ðq ,q Þ,J ðq ,q Þ,J ðq ,q Þ(cid:4)
p ðc jc Þ¼p ðc jc Þ¼y and p ðc jc Þ¼p ðc jc Þ¼1(cid:3)y : 1L 1L 2L 1L 1L 2H 1H 1H 2L 1H 1H 2H
2 1L 2L 2 1L 2H 1 2 1H 2L 2 1H 2H 1
¼½119:68,141:75,32:75,40:36(cid:4),
How does one determine the Bayesian Nash equilibrium for
thisproblemwithtwo-sidedincompleteinformation?Toanswer ½J2Lðq1L,q2LÞ,J2Lðq1L,q2HÞ,J2Hðq1H,q2LÞ,J2Hðq1H,q2HÞ(cid:4)
this question, we find the conditional expected payoffs for each ¼½78:75,108:98,30:71,44:93(cid:4):
playerfrom(10)and(11)as
Theseresultsimplythat
J^1Lðs1ðt1Þ,s2ðt2ÞÞ¼ XJ1Lðq1L,s2ðt2Þ;t2Þp1ðt2jc1LÞ ½J^1Lðq1L,q2L,q2HÞ,J^1Hðq1H,q2L,q2HÞ(cid:4)¼ð130:71,36:56Þ,
t2AT2
¼y2J1Lðq1L,q2LÞþð1(cid:3)y2ÞJ1Lðq1L,q2HÞ, ½J^2Lðq1L,q1H,q2LÞ,J^2Hðq1L,q1H,q2HÞ(cid:4)¼ð93:86,37:82Þ:
SinceinthiscaseP2’spurchasecanbelowerthanbefore,shecan
J^1Hðs1ðt1Þ,s2ðt2ÞÞ¼ XJ1Hðq1H,s2ðt2Þ;t2Þp1ðt2jc1HÞ compete better resulting in an increase in her expected profits.
t2AT2 Faced with a lower-cost competitor, P1 fares worse and his
¼y2J1Hðq1H,q2LÞþð1(cid:3)y2ÞJ1Hðq1H,q2HÞ, expectedprofitsdecrease.
Having discussed the two special cases above where one or
J^ ðs ðt Þ,s ðt ÞÞ¼ XJ ðs ðt Þ,q ;t Þp ðt jc Þ bothplayersmayhavetwotypes,itisstraightforwardtogeneral-
2L 1 1 2 2 2L 1 1 2L 1 2 1 2L
t1AT1 izethesemodelstocaseswheretheplayersmaypossessmultiple
¼y1J2Lðq1L,q2LÞþð1(cid:3)y1ÞJ2Lðq1H,q2LÞ, (more than 2) types. For example, when Player P1 has N1 types
and player P2 has one type, the Bayesian Nash equilibrium
J^ ðs ðt Þ,s ðt ÞÞ¼ XJ ðs ðt Þ,q ;t Þp ðt jc Þ solution is found by solving a system of N1þ1 nonlinear equa-
2H 1 1 2 2 2H 1 1 2H 1 2 1 2H
t1AT1 tions in N1þ1 unknowns, extending the results in (14)–(16).
Similarly,whenPlayersP1hasN typesandP2hasN types,then
¼y J ðq ,q Þþð1(cid:3)y ÞJ ðq ,q Þ, 1 2
1 2H 1L 2H 1 2H 1H 2H we solve a system of N þN nonlinear equations in the N þN
1 2 1 2
wheres ðt Þstandsforðq ,q Þands ðt Þisðq ,q Þ.Sinceeach unknowns to find the Bayesian Nash equilibrium. Naturally, as
1 1 1L 1H 2 2 2L 2H
newsvendor’s strategy set is ½0,1Þ(cid:6)½0,1Þ with moves ðq ,q Þ the number of types increases, the problem of estimating the
1L 1H
for P1 and ðq ,q Þ for P2, we follow the standard steps of typesandtheprobabilityoftheirrealizationalsoincreaseswhich
2L 2H
Table6
TheeffectofuncertaintyontheequilibriumsolutionforP2.WhenP2ispessimistic,herexpectedprofitislower,andwhensheisoptimistic,herexpectedprofitishigher.
Completeinformation c1 c2 q1 q2 J1 J2
8 5 25.38 19.55 83.63 35.91
Incompleteinformation ðc1L,c1HÞ c2 ðq1L,q1HÞ q2 ðJ1L,J1HÞ J2
P2pessimistic (6,8) 5 (36.21,25.98) 17.96 (147.96,86.16) 32.66
P2optimistic (8,10) 5 (24.73,16.54) 21.52 (80.90,39.86) 40.11