Conjectural Variations Equilibria Part I: Static Equilibria

Alain Jean-Marie Mabel Tidball [email protected] [email protected]

INRIA/LIRMM CNRS University of Montpellier 2 INRA/LAMETA Montpellier

Seminaire´ du GERAD, Montreal,´ lundi 26 avril 2004 – p. 1/33 2/33 . p – 2004 il vr a 26 lundi y ´ eal, Montr the , a decision. GERAD studies) e du v ia... wn o ha ´ ical eminaire S ium? ers y their opponents: Equilibr empir of pla incomplete or f actions their Equilibr is le Nash of ation. inter to which e function in vior iations mation (suitab in cooper ar or y nativ V beha inf dynamic pla al model model alter concept or the or? f f e will implicit al le when vior about thand Conjectur useful others possib model xplicativ a it a beha e xt to shor is is the • • As especially or A game-theoretic • • • • What A conjecture What think Conte 3/33 . p – 2004 il vr a 26 lundi ´ eal, Montr , GERAD du (CVE) actions ia ´ eminaire S ategies inter ategies str str ategies Equilibr str dynamic iations or igger f ar tr open-loop closed-loop V al with with with thands shor games games games Conjectur as ties CVE Static Definition Proper Examples Consistency Repeated Dynamic Dynamic • • • • • • •

Contents 4/33 . p – 2004 il vr a 26 lundi ´ eal, Montr , GERAD du ´ eminaire S equilibria

ariations v

conjectural

Static 5/33 . p – 2004 il vr y a y b 26 b . lundi ) b i b 2 e ´ eal, e , Montr b 1 , . e i “react” ( from GERAD du er = y will b e j ´ eminaire . S pla E viates er y y b de set pla profile , i she e if d some then ) ategy , b j i in assumed with e e i that str , , d b i i k e er ( y er j quantity: iation r y game thinks pla ar the v pla i er . j y of quantity y r al of b er y benchmar b j e off o-pla y pla ategy tw pa str some a from function that conjectur the the infinitesimal i i the V e some is viating • • j or Assume de Consider some r Consider f Preliminaries 6/33 . p – 2004 il vr a i 26 e lundi ∆ ´ eal, y Montr b , . GERAD . du )) viates b j ´ eminaire e S , de b i function e ; , i e ) she i . ( b j e c j if e ρ ∆ , i = reaction e + that ( ) b i j b j e r e y: ( , c j b i ρ = e think pla ; b i ) = e to b j ( will e c j j , ρ of: e j b i viations? conjectured led i e e ; er de i ∂ y (continued) e the ( c j is ρ pla ∂ solution logically condition is is then c j , i ρ b i e function infinitesimal initial er y This where Pla Non from with Preliminaries 7/33 . p – 2004 il vr a 26 , lundi } ´ eal, is ) c j ) Montr e , c 2 , together e c i , , e GERAD c 1 2 ; du al e , i ( e 1 ( if ´ eminaire c j S = ium ρ i Gener ) = i e a j , j e is e (GCVE) Equilibr ( E i r lem: and ∈ ium ) E c 2 e prob iations ∈ , c 1 . ) ar e 2 2 ( V , e Equilibr , 1 1 al e = conjectures ( i | ) j or ategies f iations e optimization , i str ar e iational V ( Conjectur i of the ar al V v of { pair of a max ultaneously pair with Definition: A Conjectur sim solution Definition 8/33 . p – 2004 il vr a 26 lundi ´ eal, : solution Montr , er y . the GERAD du le y pla . b ´ 0 eminaire S = each , satisfies ) j erentiab j it e or f defined , i 6 = diff e i ( i j then V ) twice j ) e c j implicitly , e is i ( conditions i e i is GCVE, ( χ ) V j j a r e = ( is i c i order + that ) e χ c 2 ) e j , e first c 1 , e i ( e conditions ( i i function Assume wing V and ) ollo the 2 f order r , 1 r the ( where Theorem: If of First 9/33 . p – 2004 il vr a is 26 has i lundi χ wn ´ eal, er , o y . Montr , mapping pla . her GERAD du )) to i no the e ( ´ eminaire functions S j of χ function. conditions , CVE, ) j e a point ( i is according order χ ed ( response it, fix response ategy best the from second str s 7→ best k as al iate viate Nash’ ) de j seen e , to i with be e e ( appropr conjectur benchmar can the are the .

ations comments: analogy incentiv CVE al By A When There called an beliefs er v • • • Se Observ 10/33 . p – 2004 il vr a al 26 lundi ´ eal, Montr , GERAD that conjectur du e ´ eminaire the S or f ategy

Outcomes str CVE . . conjectures the a 0 1 , ia is areto = = with P j j CVE ium r r game equilibr ic CVE be a y is equilibr ma Nash ) e e symmetr

Equilibria, e, ( a Nash aliz V A in es outcomes

Nash ty: ty: gener iations areto ar maximiz CVEs Proper P Proper v CVE, 11/33 . p – 2004 il vr a 26 lundi . ´ eal, are: . Montr , i . be GERAD i profit r du ce duopoly − ´ eminaire s c S The − conditions − . i not’ ) e ) j r )) e c j order e + + − Cour i + a e (3 demand. i in b first e (2 ( b − = the duopoly erse a − CVE c j v , e a

s r in ( = = i j = ore: c i V and m: e ) r j iations or e f + , ar i i theref introduced cost i v Cournot’ e V ( is the i

1: V = linear has 0 (1924) y constant solution wle function Assume The Bo With Example 12/33 . p – 2004 il vr a 26 lundi ´ eal, wing al Montr off , . y ollo GERAD f pa du . good α α the − ´ eminaire 2 1 S lic conjectur ) ) j r + e ) + pub r + i a + e )(1 b-Douglas) ( to α α proposed constant )(1 ) i − e α e v (Cob (1 − − ution with ha I ib the (1 ( I to = goods CVE ) contr j = (1995) e y , c 2 i e leads e ( i = unique

Public V c 1 a e . oluntar : 2 v

2: uction r / Dasgupta a 1 xists of < e and α constr a iations y ar function: v The Ita with There model Example 13/33 . p – 2004 il vr a 26 lundi ´ conjecture r eal, Montr 5 , GERAD 4.5 du 4 ´ eminaire S conjecture: 3.5 the 3 Pareto of 2.5 2 Nash function 1.5 in 1 off y 0.5 pa 0

(continued) the

2 −0.5 1 of 0.9 0.8 0.7 0.95 0.85 0.75 iation ar V payoff Example 14/33 . p – 2004 il vr a 26 lundi ´ eal, Montr remain... of , only ia GERAD du CVE. ´ eminaire S common consider equilibr elimination and against Nash ated should ? aised r iter mation only ers y or the inf pla mation been , or e Often, v . after . inf ha complete ationality r ategies of of str remaining Rationality incomplete objections if al er wledge But v situations ategies dominated kno Se Individual In str → Objections 15/33 . p – 2004 il vr a 26 y the lundi ma es ´ eal, v results! Montr , . le GERAD du obser ab v elberg outcome ´ er eminaire S k y y Conjectures pla an obser . Stac , all No la . à or f ? act to conjectures conjectures actions accounts the sequentiality y inter . not assumed consistent theor . is y vior are

(continued) of carefully ed”. pla . The s there ers beha repeated y if er le v pla CVE. e selecting a But Definition w opponent’ → Ho “endogeneiz Credib Both Refutability By be → Objections 16/33 . p – 2004 il vr a 26 lundi al ´ eal, Montr , best . GERAD . conjectures du al 0 ; ) Conjectur = 2 ´ eminaire S r ) , al j 1 e r , iational ( i coincide e conjectur ar ( i v j Gener V ) that of the j i e e , i reactions e in and ( j conjectures ) r c 2 e or Consistent if: f , GCVE + c 1 a e ) ( requirement j solution ium e are , a i conjectured GCVE 2 e the ( , i a i 1 is y V ategies is and Consistent = Equilibr being ) i str c 2 ) , j e ) e , of 2 ( c 1 e i e , ( χ 1 iations e pair i) ( ii) ar i V Consistency Definition: r A responses

Consistenc 16/33 . p – 2004 , il i vr a 26 er lundi y al ´ eal, pla Montr , best of GERAD . conjectures du al ; ) Conjectur 2 ´ eminaire S r . , , al ) 1 ) c j j r e iational ( e coincide , response , c i conjectur ar ) e j v ( Gener e ( i best of that the χ ( al i r reactions and = conjectures ) ) c 2 j e or e Consistent if: f , GCVE ( c 1 0 i a conjectur e χ ( requirement ium neighborhood the are conjectured GCVE 2 the , a 1 is y ategies some is and Consistent = Equilibr being ) i str in c 2 ) , j e ) e , of 2 ( c 1 e i e , then ( χ 1 iations e pair i) ( ii) ar i V Consistency Definition: r A responses

Consistenc 17/33 . p – 2004 il 1 vr a 26 lundi ´ eal, 0.8 Montr , GERAD 0.6 du ´ eminaire S 0.4 0.2 Public Good−Constant conjecture. Player 2 0 1 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

y 1 0.8 0.6 consistenc

of 0.4 0.2 Public Good−Constant conjecture. Player 1 0 1 0 0.8 0.6 0.4 0.2 Geometry 18/33 . p – 2004 il 1 1 vr a 26 lundi ´ eal, 0.8 0.8 Montr , GERAD 0.6 0.6 du ´ eminaire S 0.4 0.4 Public Good−Nash Equilibrium 0.2 0.2 0 0 1 0 (continued) 1 0 0.8 0.6 0.4 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 y 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 consistenc

of 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 Public Good−Consistant Conjectural Equilibrium 0 0 0 0 1 0 1 0 1 0 1 0 0.8 0.6 0.4 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.8 0.6 0.4 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Geometry 19/33 . p – . , 2004 il vr ) ) a j j of 26 e e , , lundi no i of i e ´ e eal, ( ( i i ii j Montr , are V V ) ) j conjectures GERAD i ultiplicity e , du e ( j ( 0 m systems f ) 00 there i ) ´ eminaire a f S j ( f ( and ) + i finds j constant f finds e solving ) ( j 0 e ) i , with find i f ( e : ( (1981) all i (1981) ij + V requires ) CVE )) j wn, i e e , ( i estigated... 0 Olsder e CVE kno ) v equations ( j . j i in f j Bresnahan V )( are , ) j be i e e ( o ( 0 0 . ) T ) i ) consistent erential j j . f solutions f e ( computing ( ( i duopoly + f solutions al, + 1 ( = w the i e e analytic solutions F In = erence-diff gener • • 0 In diff with Computing 20/33 . p – 2004 il vr a 26 lundi ´ eal, Montr , GERAD du shorthands ´ eminaire S as

interactions equilibria

dynamic ariations v

for

Conjectural 21/33 . p – 2004 il vr a 26 lundi ´ eal, Montr , GERAD good (CVE) du actions lic ia ´ eminaire S inter pub a Equilibr to dynamic ution iations or ib f ar V contr al y thands shor oluntar v Conjectur as of CVE Static inciple Pr Model Duopoly Conclusions • • • •

Contents 22/33 . p – 2004 il vr a 26 lundi static ´ eal, a Montr captures sought , y b is GERAD without du ed” or ´ iz eminaire S with optimal conjecture , . an The “summar game actions be . which ium. y the in inter ma state of y equilibr state policies orces f y k? repeated of al or stationar iations w , ar with it v amily f stationar al does game resulting the dynamics A some a inter-tempor redients • • • Ing Idea: conjectur When the Principle 23/33 . p – . 2004 ) il ) t vr a ∗ n ( i 26 e e , with lundi i . . ´ eal, . P , Montr , ∗ 1 , j = e  game ( ) GERAD t )) du ( t i ( G some ´ eminaire 6 = G S j profile − or ∀ f a , , A repeated ∗ j ∗ j good: e e )( a en t ( lic = 6 = G giv 1) 1) : pub + − − a Okamura ) t t t ( ( consider to ( j j i e e e ategies is: and if if i − str i ution ∗ i I er (2003) e 0  ib y t a θ ( Itaya function. ur pla igger =0 ∞ tr t contr X off = of of s y ) ’ · ) ( i i t e pa Okam ( max lem is i adopt e i e atic and model prob ers a y y where Pla Ita quadr The The 24/33 . p – 2004 il vr a 26 lundi ´ eal, is: Montr , . GERAD i best du ∗ − . E . ´ eminaire the ∗ θ i S e c − θ + E 1) 1 Eq., satisfies: − − r r n  ( + r i θ Nash case): 1 ∗ − + θ + iation E 1 ic 1 are ar − − v r A 1) 2 that

(continued)  − 1 2 = A (symmetr ( r = 1 2 constant r + c i ategies e 1 = yields str with ∗ i

Okamura e such CVE and a , Also Among Identification Itaya 25/33 . p – 2004 il vr a 26 lundi ´ eal, (2001) a Montr , ur . GERAD i du e , i ´ t eminaire X S d . Shimom )) = t G ( δ G G and , − ) a t y ( ) i t e ( Ita ( i ame with i e is: g U t i θ X − e ) (1991), game = ∞ G game: 0 , i ˙ Z G e ) ( the · i

dynamic ( static i e Nitzan U max i e and max

goods, considered associated e v eshtman ha F The Public 26/33 . p 0 – 2004 il = vr a 26 i G lundi U ´ eal, 0 2 Montr φ , = 1) 0 GERAD i G du − 1 U = n ( δ ´ eminaire i S G 1 + − U θ δ r + + +

(continued) i i i x x x U U U as: ame

g ite wr ium ium r equilibr G equilibr dynamic 2 φ equations iation + ar 1 Nash v Nash φ k order goods, = i loop with e first eedbac f with open CVE The Public 27/33 . p – 2004 il vr a . 26 t lundi d ´ eal,  ) Montr , ) t ( GERAD i du x costs: ( (1992) ´ eminaire A S . − ) i kner e )) ( and: t ( C i Doc e . ( adjustment − i C i Q and e − estment with ) ) is: j = v t e ( i in i , ˙ i e e e (1989) )) ( an t p game ( i ty j duopoly Q e i er , e ) k, t max ( static i e ( stoc p McCaff  a duopoly dynamic t θ a is − i e and e ∞ associated 0 Z iskill consider where Dr The Dynamic 0 28/33 . C p – 2004 = 0 i il vr a C 26  φ lundi = ´ eal,  ] 2 Montr , φ φ ) 3 r 0 φ GERAD − du C θ + + (1 ´ eminaire = S 1 + ]  φ [1 + ) + 1 E  ( [1 p p p as: ite j wr ium e ium 3 φ (continued) + r i e equilibr 2 equilibr φ equations iation + 1 ar Nash

duopoly φ v Nash k order = i loop with Q first eedbac f CVE open with The Dynamic 29/33 . p – 2004 il vr a 26 , 2 j lundi atic e ´ eal, 5 2 t a Montr , d +  GERAD j du e )) i t e ( i ´ eminaire 4 S linear-quadr a Q , ( i + al C 2 i Q . e − q i 3 e 2 )) a δ + t gener ( 2 i j + − e Q j , the e i ) c 2 t 2 Q ( a i e = ( + = i i ) i i e P ˙ e studied  1 Q t ( a θ C − + has e 0 a ∞ 0 Z = ) ) · ( j (2000) i e Q , max i e ( i P Figuières case: with Generalization 30/33 . p – 2004 il vr a ia 26 lundi , ´ eal, /c Montr , ∗ x equilibr /c ∗ − GERAD y δ du + . θ /c . ) r ´ eminaire Nash ∗ S 5 ) x ) /c a 5 4 ∗ − a y δ a + + θ q 4 + + ) a 2 4 3 δ ( r a a a 2 ( ( + − a + open-loop ) θ − − q 4 + ( ) ) c a 1 4 δ δ − a a ) and + 1 δ + 3 + a k θ a + 3 ( ( a θ ( ( − − 1 − c a δ = ) (continued) eedbac f δ = o e c + e θ ( is: = f e CVE steady-state The are: The Generalization 31/33 . p – 2004 il vr a 26 lundi ´ eal, Montr , estigated consistent GERAD static v du in ´ eminaire and S and be CVE iations: to of ar “conjectures” v ategies of games al str needs results k . ia CVE models dynamic conjectur cases eedbac f equilibr een other static uniqueness ...) een some consistent or le betw with f in iations? iations and of betw ar ar ... v v xions al al issues . alence ationalizab xistence xistence connection (r e e conne ther • • • ending The conjectur P fur Relationship Equiv conjectur Conclusion 32/33 . p – , 2004 il vr a nal ican 26 Costs: iations lic lundi 324–338. Jour ar ategic ´ . Amer eal, V ord Pub str pp al Montr The , a of Oxf , in 49, GERAD Adjustment ol. du V vision , iations”, Conjectur with y ´ ar Pro eminaire Conjectures”, S of V y y Economics al conjectures Theor of of k Duopoly 1057–1067. Theor oluntar . or V ect pp Consistent eff Conjectur 35, Economic of with Tidball, the y 377–395. of “Dynamic ol. Groundw . V M. On “Dynamic , pp nal w 934–945. Theor Models and . vie (4), (1989), Jour . pp Re Tidball, (1991), S XL . , S (5), ty M. ol. Quérou Mathematical 2004. Dynamic V er “Duopoly , 71 , 2004. N. and “A , ord. The ol. Approach”, ie ie Nitzan, Economic V , Oxf McCaff , w (1981), . y and (1992), . vie .F Game and (1924), . Economics T C Economiche Computing, Re Press Jean-Mar Jean-Mar E.J ial European , A.L. R.A. A. A. , , , y erential ersity kner wle Industr iskill, Ricerche Diff ershtman, , Scientific Goods”, Bo Bresnahan, Doc Dr F Univ a Economic of ld • • • • • Figuières Figuières or . . C W C setting Bibliograph 33/33 . p – 2004 il in vr The of a lic , 26 nal Pub lundi , Model and y y ´ eal, , estment Jour v lic 2. Montr , In echnology ategic 4(4), Theor T iations Pub oluntar GERAD V ar of Str ol. Run V du V Concept”, , al the and Conjectures w Approach”, ´ eminaire Long ersity ium S Goods”, vie Economic Aix-Marseille and in lic Re of Univ lic iations Game y Conjectur ar Pub Equilibr V Pub action Consistent ente . of ersity w , al J w Theor T Ne 371–389. Inter erential ., . Univ a of Dynamic vision Diff pp Substitutability of Game , a “A Pro Setting”, Maths ity “Dynamics “Conjectur 00A01, 50(3), ate iv Concepts ol. 153–172. Analysis Game Goods: national (2001), Pr . V Applied paper (1995), (2003), , lic K. pp . the Inter D itical M. a, Pub in king 81, Cr ur Dept. a, liques or of “Dynamic “Complementar ur ol. Repeated “A w V Pub a 329,

(continued) , . Capital”, Agents in vision Dasgupta, Okam Shimom y Nr of (2000), (2002), 51–66. (1981), . . . . Pro . C C and and and vision pp GREQAM , , G.J ate .-I. .-I. .-I. andum ulation lands Economics , Pro iv J J J 5(1), lic Pr 371–390. a, a, a, . y y y ol. Figuières Figuières Ita Ita Ita Olsder Good V the Games”, Memor Accum Heterogeneous Pub pp Finance/Finances Nether • • • • • •

Bibliograph