STRUCTURE IN LEGISLATIVE BARGAINING

Transcription

1 NOT FOR PUBICATION ONINE APPENDICES FOR STRUCTURE IN EGISATIVE BARGAINING Adan de Goot Ruz Roald Rame Athu Scham APPENDIX A: PROOF FOR PROPOSITION FOR HIGHY STRUCTURED GAME APPENDIX B: PROOFS FOR PROPOSITIONS FOR OWY STRUCTURED GAME 5 APPENDIX C: EXPERIMENTA INSTRUCTIONS

2 NOT FOR PUBICATION APPENDIX A: PROOF FOR PROPOSITION FOR HIGHY STRUCTURED GAME In ths aendx, we ovde the oof of ooston, whch chaactezes the equlbum outcome of H T (when t conveges). Snce H T s (hghly) non-convex due to the exteo dsageement ont, we cannot use standad esults and technques to deve equlba; athe t nvolves a tou de foce n backwad nducton. We also an smulatons, whch llustate (and cooboate) the esults of the ooston. In atcula, they shed some lght on what haens f the outcome does not convege. At the end of the aendx, we ovde a fgue that llustates the cyclc deendence of the outcome on a and b fo < a < and a < b < (as obtaned by smulatons). PROOF PROPOSITION Befoe we can detemne the equlbum, we need to ntoduce some notaton. Due to backwad nducton and layes havng a unque best esonse at each nfomaton set, the equlbum oosal and votng stateges only deend on how many ounds ae ahead. Hence, we wll count the ounds by the emanng numbe of ounds T t. Hence, the fst ound has = T and the last ound =. Futhemoe, ths mles that the equlbum stategy fo ound s the same fo each game H T wth T. Hence, t s meanngful to talk n geneal about the (sub)game H. The equlbum (behavoal) stategy fo laye,, secfes fo each ound T () fo the oosal stage, a obablty dstbuton ove ossble oosals : Z [,], ( ) : Z {,}, ( ). The equlbum outcome of, and () fo the votng stage, an accetance functon H can be chaactezed by the obablty dstbuton of the equlbum outcomes : Z [, ]. The contnuaton value EU E [ u ( z)] ( zu ) ( z ) s the exected utlty of su su s a obablty mass functon and has countable suot: su max su 6. T, N

3 NOT FOR PUBICATION laye of the (sub)game H. We can convenently exess EU n tems of f ( ), E [ z z R] and D E [ z z R]. (Note that D ). Defne the ndcato functon I R ( x) [ f x R, f x R], the accetance obablty ( x) ( j ( ))( k( )) and the obablty of delay P [ delay] ( x) ( x N su ) f N su. Then, we get: ( x) ( x) I ( x) P [ delay] f, R If f, then, D. Othewse: ( x) ( x) xp [ delay] f f N su \{ } D ( x) ( x) x P [ ] dela f y f N su \{ } f D (.) Snce E [u (z)] = P [z = δ] E [u (z) z = δ] + P [z δ] E [u (z) z δ] and u (z) ae lnea n z fo z [ a, b] u z z z we get that (fo su [ ab, ] ):, fom EU f ( a ) EU f ( D ) (.) EU f ( b ) A laye wll accet a oosal n ound + f and only f u ( ) EU. Ths allows us to chaacteze fo ound + (), the lagest oosal laye accets, (), the smallest oosal laye accets, and () absolute value a oosal can have fo laye to accet t: f a f f b f D, the lagest (.) D f f D Playes wll only delay f they cannot make a oosal that wll be acceted and gves them at least the contnuaton value. Playe o wll only delay n ound +

4 NOT FOR PUBICATION f D, whch s equvalent to ( f ) ( a) + f ( + D ) <. Ths can only hold f a > and f <. Hence, layes and wll neve delay f a and, by the same easonng, laye wll neve delay f b. Note that f R, s acceted n equlbum, t must be acceted by laye n ound + and, hence, ( u ) EU. If a <, then laye wll oose a (o a ) n ound T. Ths means that fo all and that she wll neve accet no oose δ. Futhemoe, f a <, laye can always oose so that she wll neve oose δ. Fnally, EU EU mles <, and hence EU f b. In the followng emma, we summase these facts and some condtons that ae easly deved. EMMA B. Fo a <, the equlbum {σ, σ, σ } s detemned by:. ( ) = fo [ ab, ] ff and ( ) = ff EU.. ( ) = fo [ ab, ] ff D and ( ) = ff EU.. ( ) = fo [ ab, ] ff ( ) and ( ) = ff EU. 4. ( ) ff D a and EU ; ( ) a ff D a; ( D ) ff ad a o D a and EU ; 5. () ff o ; ( ) ff, and ; ( ) ff, and ; ( ) ( ) ff, and ; 6. (delay) ff D and EU (only f f <); ff ( ) D o x b and EU ; ( ) b ff D b ; ( D) ff b D EU and D b. Now we ae eady to look at whethe the equlbum outcome conveges. et lm x. The obablty dstbuton s the lmt of b f t holds that

5 NOT FOR PUBICATION lm ( z) ( ) z fo all z n the suot of. As defned n secton, we say that the equlbum outcome conveges to f lm ; f ths lmt does not exst, we say that does not convege. The equlbum outcome conveges to f (), whch s equvalent to f and D. The equlbum outcome conveges to δ f ( ), whch s equvalent to f and lm su D. Fnally, t s staghtfowad that does not convege f () f does not exst o () f and o D do not exst. PROPOSITION () If a < o a = b =, then equlbum outcome conveges to () Poof: If a b < and b >, the equlbum outcome does not convege, unless 5 b, a 7 4 a b o 6,max{ b, b } a b o b, a b o b In these latte cases the outcome may convege, but neve to a sngle outcome n Z. () If a, the equlbum outcome s δ. (.a) We show that f a <, then f and D. Thoughout the oof, we wll use the followng suffcent condton fo convegence: Fo a <, f and D f f EU thee exsts a ound such that and (SC) et (SC) hold fo. EU ( a) mles that laye wll not accet no oose δ n ound +. f = mles that laye wll not delay and that ethe laye o accet n ound +. Consequently,, D D and D D. Hence, f =, = ( D D ) = < - a and D ( ) D D D. Thus, (SC) holds 4

6 NOT FOR PUBICATION fo + and by nducton fo all. As a esult, f and D m lm D m lm D. A suffcent condton fo (SC) to hold s that: m m thee s a ound ' such that ' ' ' ' ' ' ( ) D a and a and ( ) o ( a) (SC ) ' ' ' D a and a mly that ' ' D and ' D '. Hence, f ' = and ' E [ ' ' ' ' ]. If o ( a ), then ' E[ ] ( a) and ( a). Hence (SC) holds fo. In the emande of the oof we dvde the (a, b) aameteset nto egons and show that (SC) holds fo each egon. We stat by lookng at the last fou ounds. In the fnal ound, = a and =. ' If b <, then = mn{b,}, f = and ( a), such that ound satsfes (SC). So, let b. Then = δ and f =, D = = a. Hence, a < ( ) a, D a and ( ( ) b a ). Snce EU, = delay ff D ff b > (a + ). et us fst consde b (a + ) and a ½. Then = a, = and D, so that f = and ( ) 9 a < a. Hence, ound meets (SC). et us now consde b (a + ) and a > ½. In ths case, D a, so that ound satsfes (SC ) f o ( a). So let ( a). Ths means that 8a b < a and a. Futhemoe, D, and D, so that f =, a and D 9 ( + a + b) a. Hence, D D, and, so that f =, D = - = D. Fom ths, 4 (8 ab ), 7 D 4 a ) ( b and (8 b a ) ). < b (a + ) and a > ½ mly 4 D a and 4 ( a ). 8a b < a mles 4 4 D 4. Hence, ound 4 satsfes (SC). Thus (SC) holds f 5

7 NOT FOR PUBICATION b ( a ) (A) et b > (a + ) fom now on. EU fo all and laye can now delay consecutve ounds and altenatngly delay and not delay. Ths eques a caeful chaactezaton of the dynamc befoe we oceed. We wll call a set of consecutve ounds n whch laye delays a delayng sequence. We ndex these sequences by s (agan backwads), wth s = the fnal delayng sequence, s = the efnal delayng sequence etc. et R(s) be the set of ounds n the s-th delayng sequence and defne () s max R() s and s () mn R() s. Fnally, let ms () () s s () be the numbe of delayng ounds n R(s). et us look at s-. () ( s) delay mles EU and ( s) f ( s). As laye accets δ n s () -, she also accets and ( s ). Snce D ( s) ( s), ths means that ( s) D D. We oceed to ounds ( s) ( s) R(s). Snce laye delays, ( s) ( s) D and mn{, }. ( s) ( s) As D and ( s) ( s) f, by (.) and emma B t must be that f <, ( s) D and EU fo all R( s). Futhemoe: If D, then: D f a D f af D D f b D f b f D In atcula, (.4) holds fo = s ( ),.., s ( ). (.4) Moeove, as laye delays n ounds R( s) and f ( s) : f f R() s (.5) m ( s) m f m fo m,,,.., m ( s ) (.6) ( s) 4 ( s) Fom (.4) we get = a D and t tuns out that ( s) () s fo s () s (). Fo s = t s smle. Suose = a and D = - = a. Then 6

8 NOT FOR PUBICATION mmedately = a. Futhemoe, due to the symmety D + = - + = a. Snce () D D = = a and () a, by nducton t follows that a fo () (). Fo s >, we need to assume that ( s) and D ( s) a and justfy t late. Suose, D a and delay. Hence, and, usng (.), D = ( ) f D f = ( f ) a f f D. Substtutng ths tem and usng (.5), we get that = a ( f f ) + f D - f D =. Hence,. Futhemoe, usng the same substtutons, we get D D f < and D D a. Fnally, D ( a )( ) D D. Hence, as s () s (). delay R() s, ( s) fo Usng (.4) and (.6), we get the followng esults fo m =,,.., m(s): ( s) m ( s) m ( s) m a ( ) s m ( ) ( s) m ( s) m ( s) m a D ( ) s m ( ) (.7) ( ) ( s) m ( s) m s m ab D () m s ( s) m ( s) m ( s) m ab D ( s m ) Snce, laye only delays n ound ff EU and >, fom (.7) we get that m(s) = ( s) m mn{ m : } ln( ab) ln( R = celng ( s)) ln(). ( s) m( s) ( s) m( s) Equvalently, snce and, we get: In ound () s () s m() s ( s) () D ( ) ( ) ms ms a ( s) ba ( s) s ( s) EU (.8) laye wll not delay. Snce and ( s) D a, () () () ( s) ( s) satsfes SC f ( a) (.9) ( 4 ) D D 9 a. Hence, by (.9) () + satsfes (SC ) f 6 a (B) 7

9 NOT FOR PUBICATION 6 et a.it tuns out that f () s does not satsfy (SC), then () s ( s ). et () s + not satsfy (SC). In ths case EU and ( s) ( s) ( s). Consequently,, ( s) ( s) and ( s) ( s) D D ( s) ab () s 6 ms ( ) ( s ( s) D ) ( ) (usng (.7)). Thus, f, ( s) EU and ( s) D ( s) ( s) ( s). Usng (.4), we get ab 9 ms. Futhemoe, snce ( s) b6 ( s) ( ) () s = 4 ( s) a D and, by D a, ( s) ( s) () a (.8), ( ) s ms a b. Hence, ( s) ab 6 b 6 a 9 a b. Snce b > (a + ) and a, a >. 9 ( s) Thus, ( s) delay and () s ( s ). As a consequence, ( s ) = 4 ( s ) a D = 4 ( s) a D : a b ( s) a ( s) (.) ( ) 9 ms We conclude ou chaactezaton of the delayng sequences by showng we can s ndeed assume ( ) and D ( s) 6 a fo s >. Snce ( s) a and a, ( s) D ( a) ( s) a. Showng ( s) eques some wok. et ( s) o s =. Usng (.), ab () s 6( a ( ) ) 8 ms. Snce ( s) ( s) () s does not satsfy (SC ), ( s) ( a) and, usng (.7), ths 6 6 a ( ) mles s. Futhemoe, (.) mles that ms ( ) () s a a fo s >. b Hence, ( s) (8 9 a a( b7)) 7(ba) < (as b > (a + )). Snce n () atcula, by nducton t follows that ( s) fo all s >. 8

10 NOT FOR PUBICATION We oceed by dvdng the aametelane not coveed by (A) and (B) accodng to m(), the numbe of ounds laye delays n the fst delayng cycle, and oof that (SC) holds fo some () s. By () = a and (.8) By (.9), () satsfes (SC ) f m() () a ab a m a a b a < ( - a) f () m() (.) b (5 ) a m() m() (C) Now, let () not satsfy (SC) and (A) - (C) not hold. Usng () = a and a b 4 (.), we get () = () 9 a. Hence, by (.7), m a7b a 9. Ths s ostve ff: () m() 4 () m b a (D) ( 5 m() ) 7 a Ths means that f (D) holds m() > m() and () () m (). Usng (.7) and the uebound fo b n (.), we get Hence, f (D) s met () satsfes (SC ). () m () ab 9 a () m ( a). Fnally, let () not satsfy (SC) and (A) - (D) not hold. As long as m(s) = m(), fom (.) we get a b ( s ) a ( s) () 9 m (.) The unque steady state of ths dffeence equaton s a b a () 8 m, whch s a global attacto wth a monotonc dynamc snce d( s). Usng the d () s 6 uebound fo b n (.), m() and a, we get that ( 7 a) a (). Hence, ( s) deceases monontoncally to. Suose m(s) = m() fo all delayng sequences. Usng the ghthandsde of (C) as lowebound fo b and a 9

11 NOT FOR PUBICATION b a <, we get () m 9( a) () 4 m. Hence, f m(s) = m() fo all s, thee exsts an ŝ such that ( sˆ ) ( sˆ ) b () sˆ () m and ŝ meets (SC ). m(s) s nceasng n s, because ( s) s deceasng n s and, by (.8), m(s) s deceasng n (s). Ths means that f m(s) s not equal to m() fo all s, thee exsts ab an s such that m(s ) > m(s - ) = m(). Futhemoe, () s a () m (fo s > ). Usng (.7), ths mles that ( s') ( s') and ( s') satsfes (SC) f ( ab) 4 a m(). As a consequence, 6 (8 4 m() ) b a a (E) 4 Hence, f (E) holds, ethe m(s) = m() fo all s o not, both of whch mly that (SC) holds fo some () s. Futhemoe, the ghthandsde of (E) mnus the m ghthandsde of (D) s a 8 and ths s ostve f a and m(). Hence, snce (A) (D) do not hold, (E) must hold. In concluson, fo each ( ab, ) [,) [ a, ) thee exsts some ound N that satsfes (SC) and, hence, the outcome conveges to as nceases. (.b) We show that f a = b =, then lm f and lm D. In ound,, and. In ound >, D, and D, wth f and lm D.. Consequently, lm f and D D D () We show that f a < and b >, then f and f, o D does not exst, excet f a b 7 5 o b, a o 8 b a 5 b 4 o b 7 4 4, 7 7

12 NOT FOR PUBICATION,max{ b, b } a b. In these latte case, the outcome may convege but neve to a sngle outcome n Z. Poof: et a < and b >. Fst, we show that f the outcome conveges thee exsts an such that delay fo all >. Snce EU and D, and fo all. Suose thee exsts an such that delay fo all. Ths mles that EU, and D a fo all and hence by (.) that f = '. Futhemoe, usng (.), (.4), and the logc behnd (.7), we get that and, hence, D = ( ) D ( ) ' fo all. Now, delay only f D, whch mles by (.4) that ( f )(b ) f D > fo all. Howeve, ths s not ossble, snce f = and D. ' ' Hence, thee does not exst an such that D ) mles the ooste holds: delay fo all. Convegence (of thee exsts an such that delay fo all (.) Second, thee can be no convegence to δ as fo all. Thd, thee can be no convegence to. lm f and (.) would mly that thee exsts an such that f fo. Ths means that D, and D fo all +. Consequently, = and, hence, EU and EU < fo +. Howeve, f ths s the case, contadctng f fo +. Fnally, we show thee can be no convegence to anythng else then o δ, save fo fou excetons. Suose that f, and D exst, but f and D. Now, f =, o. We have seen above that f = s not ossble. et f =. Ths means that thee exsts a ound such that f =, D a and, thus, D a 5 fo. Suose mn{, } fo. Ths mles that - = D - and <, such that. Futhemoe, D ( a ) D. Convegence

13 NOT FOR PUBICATION mles that lm D D and solvng fo D D yelds D = = a. Howeve, snce a <, D = a > a 5 and D a 5 cannot hold fo all. Hence, mn{, } and a smla easonng (wth a b < a 5) shows that max{, }. Thus, let ( ) ( ) fo. Ths means that =, and, hence, a = b fo. Futhemoe, ( ) Hence, = and a. D a 5 eques 7 ab. D a. et, ultmately, f =. Ths means that thee exsts a ound such that f =, - a, a < D b, D, and mles fo. In atcula, ths a and a D b (.4) To begn, suose. Now, = D and D = D 6 D fo >. Solvng fo D D yelds D = =. As 4 4, and eques. Togethe wth (.4), ths mles that b and 5 a. Suose now that 4 fo. Hence, = D and D = ( D ). Solvng D 6a fo D D gves D = a =. Howeve, ths s not ossble due to (.4), snce a > a. To contnue, suose ( ) ( ) fo. = (4 ) ba mles ( a b) 4 fo. Futhemoe, D D 4 4 a b4d and D 4 4 ba D. Solvng fo = - and D = D - and usng 4 4 a b ), we get that = ( 4 8 b and 5 4 a b (a b) and = 4 D 8 (a b). (.4) mles that The last ossblty s. Thus, < and a b) fo. Futhemoe, D = D 6 b D 6 b D, = D. Solvng fo = - and D = D -, we get = 4 ( b 5

14 NOT FOR PUBICATION D 5 and = max{, } b b a b b. (.4) and a b) mly that 4 ( b 7 and In concluson, f a < and b >, then a necessay (but not necessaly suffcent) condton fo convegence s that ethe of the followng holds: ( ) a b wth f,, D ( a) ( ) b, a wth f, D ( ) b, a b wth f, ( a b), D ( a b); ( v) b,max{ b, b } a b wth f, b, D b. 5 (Note that these fou egons coves a vey small at of the aamete set.) (.5) () If a, then f and lm su D Poof: It s mmedate that. fo all and. Hence, f = and D = fo all.

15 NOT FOR PUBICATION CYCES To llustate the cyclc deendence of the equlbum outcome on T when the coe s emty, we ovde below the smulaton esults fo < a < and a < b <. The colo of the aea ndcates the eod of the cycle. Whte egons ndcate that thee s a steady state. The dake the colo of the aea, the hghe the eod of the cycle. The dakest colo ndcates the eod s equal o hghe than.. b a 4

16 NOT FOR PUBICATION APPENDIX B: PROOFS FOR PRORPOSITIONS FOR OWY STRUCTURED GAME In ths aendx, we ovde the oof of oostons and n secton. PROOF OF PROPOSITION To ove that some ofle movement: s well-defned, we need to show how the game oceeds gven and some hstoy h τ. We fst defne the fst moment of DEFINITION. Gven and h H and let R ( {τ t T: h ) ( ht( h)) ( ht ( h )) }. () ( h ) s the fst moment of movement of laye. If R( ), then ( h h ) T and laye would not move at any h( h ) h ( h ). If R( h ), then ( h ) m n R( h ). () We defne the t T fst moment of movement ( h ) mn N { ( h )}. If R, mn R must exst, because othewse (S) would not hold fo hnf ( h ). Now we can defne a functon γ that etuns a hstoy h as a functon of any unesolved hstoy h s. The functon detemnes whethe the (absence of a) fst moment of acton dectly leads to a esolved hstoy. If ths s not the case, t etuns anothe unesolved hstoy wth ( h) ( h) DEFINITION. Defne : H H H, (, ) h, as follows: - h ( h ) h', wth ( h ). - If T, then h h ( h ) H. T - If T, then t ( h) ( h( h )), j t, ( h) - h' - h' h H f T T o ( h ) aj fo some,j H and ( h') f T and ( h ) a, j j R Now, t s staghtfowad to show that that T s well-defned. 5

17 NOT FOR PUBICATION PROPOSITION. The game T s a well-defned mang G: H H,( h, ) h T T Poof: Consde and h H. ales γ teatvely. It stats wth ( h ) h. If k k ( ) H, then ( ) ( k ( )) k h h. If ( h ) H, then the h k ocedue stos and h ( h ). Because ( ( h )) ( ) and T / s fnte, ths ocedue wll always etun a esolved hstoy. T h PROOF OF PROPOSITION By we denote the vecto (,, ) and by - we denote the vecto (, ). Fo convenence, we set u ( ). DEFINITION. By ˆ / zˆ we denote the stategy ofle such that the followng - hold.. Fo each actve hstoy h H T wth τ > T ρ () ( h ) aj ff (a) ( u ( h )) u ( ) and (b) j j j () ( ) ( h ) zˆ, j u ( zˆ ) u ( ) and u ( ( h )) u ( zˆ ), o (c) u ( ( h )) u ( ( h ))o j k k ( h ) ˆ z and u( j ( h )) u( k ( h )) u ( ) and ethe ( h ) k ( h ) o j ( h ). ff ( ),, h h a a a. Fo each actve hstoy h H T wth T ρ < τ T ρ () ( h ) aj ff (a) u ( ( h )) u ( zˆ) and u ( ( h )) u ( ( h )) o (b) j ( h ) zˆ and u ( ( h )) u ( z ˆ) o (c) j k j k j k ( h ) zˆ j and u( j ( h )) u ( ( )) ( ˆ ) k h u z and ethe ( h ) ( h ) o j k () j ( h ). ( ) ˆ h ff ( h ) a, a, a. z ˆ / zˆ zˆ f ( h ) ˆ and τ T ρ. h Such ofles have a secal oety: 6

18 NOT FOR PUBICATION EMMA. If ˆ / zˆ s a SPE of some subgame wth T\ h h HT, then t s a SPE fo any subgame. T\ hˆ Poof: et ˆ / zˆ be a SPE of and let ˆ be a subgame of. Fo all T\ h ĥ wth (h) ˆ > T ρ, t s mmedate fom the defnton of ˆ / zˆ that s a SPE of T\ h T T\hˆ. Hence, consde some ĥ wth ( ĥ ) T ρ and let us look at whethe thee exsts some such that U ( ; hˆ ) U ( ; ˆ h ). If layes j and k adhee to, then ' ( h ) ˆ fo a ll h hˆ. At ĥ each laye accets accodng to any oosal yeldng a hghe ayoff than devaton s that one of the followng holds: u ( z ˆ). Hence, a necessay condton fo a oftable. a subhstoy h hˆ exsts such that laye does not accet unde the most attactve oosal yeldng he a hghe ayoff than u ( z ˆ) (.e. j k : ( h ) aj, ( h), u( j ( h )) u( zˆ) and u ( ( h) ) u( k ( h )).) j. a subhstoy h hˆ exsts whee she can make a devatng oosal wth a hghe ayoff than ẑ that wll be acceted n the next actve hstoy h h. (.e. fo some actve hstoy h, z : ( h ) z, u ˆ () z u() z and fo some j, j( h ) a fo h h, ( h ) zand ( h ) ˆ.). u ( ) u ( ẑ) and a subhstoy h hˆ exsts whee she can devate by movng to be slent such that at the next actve hstoy no oosal s acceted. (.e. ( ), (,, a ) j h k( h) a a fo h h wth ( h ) qand ( h ) ˆ.) Fom the defnton of ˆ / zˆ t follows that h a h a ( ) j ff ( T) j fo h wth T T h ( h T) ( h ). Hence, f ethe of afoementoned - would 7

19 NOT FOR PUBICATION hold, then laye could also oftably devate at ethe h o h and would not be a SPE of. T \ h Hence, no laye cannot oftably devate fom, and s a SPE of. T T\ hˆ We ae now eady to chaacteze the equlbum outcomes of T. PROPOSITION. The set of SPE outcomes s equal to [ cb, ] whee cmn{ a, max{ b, b}}. Poof: We fst show by constucton that [ cb, ] T fo any wth T ρ, ae SPE outcomes of f T ρ. Fo z, we smly need to obseve that (,,) / s a SPE of any T\ h and hence. Fo z [ a,) consde the followng ofle: σ s equal to T ˆ/ zˆ (,,)/, excet that ( h ) (, z, z) and ( h ) ( h ) (, a, a) wth ( h ) (, z, ) and ( h ) (,, z). Now, (,,) / s a SPE of any subgame T T and h s the only actve subhstoy of h and h. Hence, t only emans to be shown that no laye can oftably change stateges at h, h and h. At h, laye wll obtan he maxmal ayoff. Futhemoe, at h laye no can oftably devate: nethe of them can accet the othe s oosal and, whateve they oose at h, laye wll accet a at h ( h ) gven that the othe ooses a. By the same easonng, at h no laye can oftably devate. Fnally, no laye can oftably devate at h. If laye moves away fom, the outcome wll be, whch s wose fo he than z. Playes and cannot do bette by oosng anythng else; n atcula, even f - a < z < oosng δ at t= s not attactve fo them, because that wll tgge the subgame n whch a s the outcome (athe than laye accetng δ). Hence, ˆ / zˆ s a SPE of T. In a smla way, SPE of T can be constucted that suot z(, b] as outcome. A SPE of that suots δ s ˆ / zˆ (,, )/, whch s obvously a SPE of any T 8

20 NOT FOR PUBICATION. Fnally, a SPE that suot z[max{ b, b}, a) T\ h T (f b < b < -a) s the followng ofle: σ s equal to ˆ / zˆ (,, )/, excet that ( h ) ( z, z, ) and ( ) ( bb,, ) fo all h wth ( h ) z h no laye can oftably devate fom σ at any h o. o ( h ) z. It s easly vefed that Second, we show that all onts n R outsde of [ cb, ] cannot be equlbum outcomes. Suose h s a SPE wth outcome zˆ R\[ c, b]. In ths case, laye can n equlbum neve accet x at a hstoy h, because then ethe laye o could oftably devate by oosng at t ( h ). If s oosed, namely, then n equlbum ethe laye wll accet ths, o t wll tgge a subgame n whch s the outcome unde. Playe wll n equlbum neve accet x wth x > a, because then laye could oftably devate by oosng a at t t ( h ) mmedately ules out the ossblty that z(, b) ( b, ). If by the same easonng. Smlaly, laye wll neve accet an x wth x > b n equlbum. Ths ẑ [-b, mn{ a, max{ b, b }), then t must be acceted by laye. Howeve, laye could then oftably devate by at no hstoy accetng ẑ. Snce layes and wll neve accet a oosal outsde [-a, a] n equlbum, the outcome would always be bette than ẑ fo laye. 9

21 NOT FOR PUBICATION APPENDIX C: EXPERIMENTA INSTRUCTIONS We esent the Englsh tanslaton of the ognal nstuctons n Dutch fo both teatments. INSTRUCTIONS OW TREATMENT Instuctons You wll ntally have ffteen mnutes to go though these nstuctons. When tme s u, we wll ask whethe thee s anyone who would lke some moe tme. In case you need moe tme, lease ase you hand and we wll smly gve you the exta tme you need. Intoducton In a moment you wll atcate n a decson makng exement. The nstuctons ae smle. If you follow them caefully, you can ean a consdeable amount of money. You eanngs wll be ad to you ndvdually at the end of the sesson and seaately fom the othe atcants. You have aleady eceved fve euos fo showng u. In addton, you can ean moe money dung the exement. In the exement the cuency s fancs. At the end of the sesson, fancs wll be changed nto euos. The exchange ate s euo fo each fancs. In ths exement you can also lose money. To event that you eanngs become negatve, you wll eceve at the begnnng of the exement 75 fancs exta. In the unlkely stuaton that you fnal eanngs wll be negatve, you eanngs wll be zeo (but you kee the fve euos fo showng u.) You decsons wll eman anonymous. They wll not be lnked n any way to you name. Othe atcants cannot ossbly fgue out whch decsons you have made. You ae not allowed to talk to othe atcants o communcate wth them n any othe way. If you have a queston, lease ase you hand.

22 NOT FOR PUBICATION Peods and Gous The exement conssts of 4 eods, each of whch wll be caed out n gous of thee layes. At the begnnng of each eod, atcants wll agan be andomly dvded nto gous of thee. The chances that you wll be wth any othe atcant n the same gou fo two consecutve eods ae theefoe vey small. Choces and Eanngs In each eod, you gou of thee atcants negotates about choosng a numbe. The chosen numbe detemnes the eanngs of each of the thee atcants fo that eod. The gou can choose any ntege between and. The gou can also choose not to detemne any numbe (the no numbe oton). Hence, the numbe chosen by the gou detemnes the eanngs fo each membe of the gou. These eanngs ae dffeent e membe nevetheless. How much a laye eans deends, n addton of the chosen numbe, also on he deal value. Each laye n a gou eceves an deal and unque value between and. The eanngs fo a laye ncease as the outcome les close to ths deal value. If the outcome s exactly equal to the deal value of a laye, then ths laye eceves the maxmum eanngs of fancs. The dffeence between the deal value and outcome (f any) deceases the eanngs by the same amount n fancs. Fo nstance, suose you deal value n a cetan ound s 54. Then you eceve fancs f the outcome of the eod s 54, 9 f the outcome s 5 o 55, 8 f the outcome s 5 o 56 etc. You eanngs may also be negatve. If the gou, fo nstance, chooses the numbe, then wth an deal value of 54, you eanngs wll be equal to -4. The outcome of a eod can also be that the gou eaches no ageement. Hence, one chooses no numbe. In ths case each membe of the gou eceves fancs. Dung a ound, layes ae dentfed by a lette: A, B and C. These ae based on the deal value: the laye wth the lowest deal value s A and the laye wth the hghest deal value s C. Fo nstance, suose the deal values of the thee layes ae 6, 54 and 86. Then the laye wth deal value 6 s laye A, the laye wth deal value 54 s laye B and, fnally, the laye wth deal value 56 s laye C.

23 NOT FOR PUBICATION The negotatons The gou negotatons on how to choose a numbe consst of seveal stes. Fst, we gve an ovevew. Aftewads, we dscuss the seaate stes one at a tme.. Befoe the negotatons stat, each laye can send a vate message to each othe membe of the gou. A message s a suggeston fo the numbe to choose. Each message fom one laye to anothe emans secet fo the thd laye.. Then, thee wll be.5 mnutes dung whch atcants can make and accet oosals. A oosal s a numbe between and o a oosal to end the negotatons.. As soon as a oosal s acceted by a laye othe than the oose, the negotatons end. The acceted oosal s the gou s choce fo that eod. 4. The eod also ends f afte two and half mnutes no oosal has been acceted. The outcome s then no numbe and all layes ean fancs. Infomaton sceen The fst sceen that you wll see n a eod, wll show whch laye you ae (A, B o C) and the deal values of you and you gou membes. You own lette s maked n ed. If you ae eady to oceed, befoe the tme has elased, you can ess the OK-button. Sendng and ecevng messages Subsequently, you wll be able to send a message to each of you two gou membes and they wll be able to send a message to you.

24 NOT FOR PUBICATION A message s ethe an ntege between and o the wod end. A numbe s a suggeston fo the gou choce. Wth end you tell the two layes that you do not want to negotate (and theefoe have eanngs ). You can also choose to send no message by not fllng out anythng o tyng the sace ba. To send a message, you fll out a numbe o end n one o both cells and you ess OK. Attenton: suggestons you send as a message ae not ut to a vote and wll only be seen by the laye who eceves the message. You eceve seconds to send messages. If you do not fll out anythng and ess OK wthn ths tme, then no message wll be sent. The othe layes wll only see a sace at the cell n ths case. Afte the seconds have elased, you wll see the messages that the othe layes sent to you. You wll NOT see what the othe layes sent to each othe. Makng and accetng oosals You ae then eady to make and accet oosals. In ths hase you wll see n the to-left cone of you sceen all the necessay nfomaton (you dentty, the messages, the deal values). At the end of these nstuctons, we wll show you the ente sceen lay out.

25 NOT FOR PUBICATION As a gou you wll have two and a half mnutes (5 seconds) to accet a oosal (o not accet one). A oosal can once agan be any ntege between and o the wod end. Dung ths hase, you can do thee thngs: make you own oosal, evse you own oosal o accet a oosal by anothe laye. To make a oosal, you fll out the numbe o wod you want to oose and ess on the OK button. Ths oosal wll become mmedately vsble to the othe layes n the lst outstandng oosals. Each of the othe two layes can accet a oosal you make. To evse you oosal, you smly make anothe oosal. Ths must be dffeent fom the evous oosal. The old oosal dsaeas fom the lst Outstandng Poosals (but, as we shall see late, t wll eman n the lst Made oosals on the left of you sceen). The new oosal elaces the old one n the lst Outstandng Poosals If one of the othe layes has made a oosal, then you can accet a oosal. You do ths by clckng on the oosal you want to accet n the lst Oen Poosals and ess the button Accet ths Poosal. As soon as a oosal has been acceted by a laye, the eod ends. The choce of the gou fo ths eod s then the acceted oosal. If no oosal s acceted wthn the two and a half mnutes, then the gou chooses no numbe and all layes eceve onts. 4

26 NOT FOR PUBICATION Results At the end of each eod, you wll get to see the outcome and the coesondng eanngs. Sceens Thee s a lot of nfomaton you can use whle you ae makng you choces, You can fnd: - the laye you ae - the deal values of each laye - the messages you sent and eceved - the oosals that have been ejected - the outcomes of evous eods At Pevous Peods, you can fnd the outcomes of evous eods, togethe wth the deal values of the laye and, between backets the eanngs. The wod You befoe the value and ayment ndcates whch laye you wee. At the fa-left cone below you see n ed the total amount of onts (Eanngs) that you have made acoss ounds. Because you eceved 75 fancs at begnnng, the counte stats at 75. Dvde the fnal scoe by to detemne you eanngs n euos. All nfomaton about evous eods s shown togethe on the left sde of the sceen. On the ght sde of the sceen you wll fnd new nfomaton and/o what acton you have to take. On to, the deal values of all layes ae dslayed. Fnally, you can fnd n the fa-left cone below a hel box wth shot descton of what you have to do. 5

27 NOT FOR PUBICATION. 6

28 NOT FOR PUBICATION INSTRUCTIONS HIGHTREATMENT Instuctons You wll ntally have ffteen mnutes to go though these nstuctons. When tme s u, we wll ask whethe thee s anyone who would lke some moe tme. In case you need moe tme, lease ase you hand and we wll smly gve you the exta tme you need. Intoducton In a moment you wll atcate n a decson makng exement. The nstuctons ae smle. If you follow them caefully, you can ean a consdeable amount of money. You eanngs wll be ad to you ndvdually at the end of the sesson and seaately fom the othe atcants. You have aleady eceved fve euos fo showng u. In addton, you can ean moe money dung the exement. In the exement the cuency s fancs. At the end of the sesson, fancs wll be changed nto euos. The exchange ate s euo fo each fancs. In ths exement you can also lose money. To event that you eanngs become negatve, you wll eceve at the begnnng of the exement 75 fancs exta. In the unlkely stuaton that you fnal eanngs wll be negatve, you eanngs wll be zeo (but you kee the fve euos fo showng u.) You decsons wll eman anonymous. They wll not be lnked n any way to you name. Othe atcants cannot ossbly fgue out whch decsons you have made. You ae not allowed to talk to othe atcants o communcate wth them n any othe way. If you have a queston, lease ase you hand. Peods and Gous The exement conssts of 4 eods, each of whch wll be caed out n gous of thee layes. At the begnnng of each eod, atcants wll agan be andomly dvded nto gous of thee. The chances that you wll be wth any othe atcant n the same gou fo two consecutve eods ae theefoe vey small. 7

29 NOT FOR PUBICATION Choces and Eanngs In each eod, you gou of thee atcants negotates about choosng a numbe. The chosen numbe detemnes the eanngs of each of the thee atcants fo that eod. The gou can choose any ntege between and. The gou can also choose not to detemne any numbe (the no numbe oton). Hence, the numbe chosen by the gou detemnes the eanngs fo each membe of the gou. These eanngs ae dffeent e membe nevetheless. How much a laye eans deends, n addton of the chosen numbe, also on he deal value. Each laye n a gou eceves an deal and unque value between and. The eanngs fo a laye ncease as the outcome les close to ths deal value. If the outcome s exactly equal to the deal value of a laye, then ths laye eceves the maxmum eanngs of fancs. The dffeence between the deal value and outcome (f any) deceases the eanngs by the same amount n fancs. Fo nstance, suose you deal value n a cetan ound s 54. Then you eceve fancs f the outcome of the eod s 54, 9 f the outcome s 5 o 55, 8 f the outcome s 5 o 56 etc. You eanngs may also be negatve. If the gou, fo nstance, chooses the numbe, then wth an deal value of 54, you eanngs wll be equal to -4. The outcome of a eod can also be that the gou eaches no ageement. Hence, one chooses no numbe. In ths case each membe of the gou eceves fancs. Dung a ound, layes ae dentfed by a lette: A, B and C. These ae based on the deal value: the laye wth the lowest deal value s A and the laye wth the hghest deal value s C. Fo nstance, suose the deal values of the thee layes ae 6, 54 and 86. Then the laye wth deal value 6 s laye A, the laye wth deal value 54 s laye B and, fnally, the laye wth deal value 56 s laye C. The negotatons The gou negotatons to choose a numbe consst of seveal stes. Fst, we gve an ovevew. Aftewads, we wll dscuss the seaate stes one at a tme.. Befoe the negotatons stat, each laye can send a seaate message to each othe membe of the gou. A message s a 8

30 NOT FOR PUBICATION suggeston fo the numbe the gou can choose. Each message fom one laye to anothe emans secet fo the thd laye.. Next, at most ounds follow wth makng oosals and votng.. Dung each ound, each of the thee atcants makes a oosals. Ths oosal can be any numbe between and o a oosal to end the negotatons. Subsequently, one of the thee oosals s andomly chosen to be ut to a vote. The othe two atcants can then vote Fo o Aganst the chosen oosal (the laye who made the chosen oosal automatcally votes n favo). 4. If one of these two atcants votes Fo, then the oosal s acceted and the eod ends. If both atcants vote Aganst, then the oosal s ejected and thee wll be a next ound of makng oosals and votng. Ths can contnue untl nne oosals have been ejected; f also the tenth oosal s ejected, then the eod ends and the outcome s no numbe. 5. If a oosal to end the negotatons s acceted, then the outcome s no numbe and, consequently, all layes eceve fancs. If a oosed numbe s acceted, then ths numbe s the choce of the gou fo that eod. Infomaton sceen The fst sceen that you wll see n a eod, wll show whch laye you ae (A, B o C) and the deal values of you and you gou membes. You own lette s maked n ed. If you ae eady to oceed, befoe the tme has elased, you can ess the OK-button. Sendng and ecevng messages Subsequently, you wll be able to send a message to each of you two gou membes and they wll be able to send a message to you. 9

31 NOT FOR PUBICATION A message s ethe an ntege between and o the wod end. A numbe s a suggeston fo the gou choce. Wth end you tell the two layes that you do not want to negotate (and theefoe have eanngs ). You can also choose to send no message by not fllng out anythng o tyng the sace ba. To send a message, you fll out a numbe o end n one o both cells and you ess OK. Attenton: suggestons you send as a message ae not ut to a vote and wll only be seen by the laye who eceves the message. You eceve seconds to send messages. If you do not fll out anythng and ess OK wthn ths tme, then no message wll be sent. The othe layes wll only see a sace at the cell n ths case. Afte the seconds have elased, you wll see the messages that the othe layes sent to you. You wll NOT see what the othe layes sent to each othe. Makng a oosal You ae then eady to make and accet oosals. In ths hase you wll see n the to-left cone of you sceen all the necessay nfomaton (you dentty, the messages, the deal values). At the end of these nstuctons, we wll show you the ente sceen lay out.

32 NOT FOR PUBICATION A oosal can once agan be any ntege between o o the wod end. To make a oosal, you fll out ths numbe o wod and ess OK. To hel you calculate quckly whch ayments belong to whch oosal, you also have a eanngscalculato at you dsosal. If you fll out any numbe and ess Calculate then the eanngs wll aea that all membes would eceve should that oosal be acceted. Ths devce s only meant to hel you. Nothng that you tye thee, wll be seen by the othe layes. ATTENTION: In each ounds, eveybody flls out a oosal. Howeve, only one of these oosals s (andomly) chosen. Ths oosal wll be evealed to the othes and be ut to a vote. You wll eceve 4 seconds to make you oosal. If you do not tye n anythng wthn ths tme, then wll be you oosal.

33 NOT FOR PUBICATION Votng Afte eveybody has made a oosal, t wll be evealed whose oosal has been chosen. Moeove, the ayments eveyone would eceve f ths oosal would be acceted ae also shown. Next, the oosal wll be ut to a vote. The laye who made the oosal, automatcally votes Fo and does not ess any button. The othe membes can vote by smly essng Fo o Aganst. If at least one of the two votes s Fo, then the oosal s acceted and t wll be the outcome of that eod. The gou has then made a decson and the eod ends. If both vote Aganst, then the oosal s ejected and you wll oceed to a next ound of oosng and votng. Ths can contnue untl nne oosals have been ejected. If the tenth oosal s also ejected, then the gou was not able to each a decson and the eod ends. In ths case, the outcome s no numbe. Results At the end of each ound, you wll see how each laye voted, whethe the oosal has been acceted and whethe o not you wll go to a next ound. At the end of each eod, you wll see the outcome and you coesondng eanngs.

34 NOT FOR PUBICATION Sceens Thee s a lot of nfomaton you can use whle you ae makng you choces, You can fnd: - the laye you ae - the deal values of each laye - the messages you sent and eceved - the oosals that have been ejected - the outcomes of evous eods At Pevous Peods, you can fnd the outcomes of evous eods, togethe wth the deal values of the laye and, between backets the eanngs. The wod You befoe the value and ayment ndcates whch laye you wee. At the fa-left cone below you see n ed the total amount of onts (Eanngs) that you have made acoss ounds. Because you eceved 75 fancs at begnnng, the counte stats at 75. Dvde the fnal scoe by to detemne you eanngs n euos. All nfomaton about evous eods s shown togethe on the left sde of the sceen. On the ght sde of the sceen you wll fnd new nfomaton and/o what acton you have to take. On to, the deal values of all layes ae dslayed. Fnally, you can fnd n the fa-left cone below a hel box wth shot descton of what you have to do.

35 NOT FOR PUBICATION 4