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On sequential negotiation procedures
Reglonal
Science
and Urban
Economics
ON SEQUENTIAL
20 (1990) 537-556
NEGOTIATION
Optimal Negotiation Yasushl
North-Holland
PROCEDURES
Orders and Land Prices* ASAMI
Unrterslty of Tokyo, Tokyo, Japan
Aklhlro Burldmg Research Institute, Received
TERAKI
Mzmstry of Constructmn, Tsukuha, Japan
May 1988, Iinal verS,on received Aprd
1990
Models of land procurement by a developer are dnaiyzed The developer IS assumed to require at least a certam Size of contiguous land to obtam a prolit, wtuch he tries to procure by sequential negotlatlon procedures The results suggest a ‘maxImum principle of negotiation’, namely that an optlmal order of negotiation should always satisfy the condition that regardless of whether or not any given negotlatlon fads, the remammg order contmues to be optimal for the rest of negotiations It may be optimal for the developer to procure land unit wluch IS never used m the development, so that he can strengthen tus bargaining position If the Size of landowners’ land units are different and all units are required, the developer will choose to negotiate the owner with the largest land unit to maxlmlze his payoff
1. Introduction The properties of price or rent profiles m markets with finitely many participants have recently analyzed by several researchers [cf Eckart (1985), Asaml (1987,1988)] Examples of such markets include situations where a few large developers are Involved m the procurement of land Consider a simple problem, for example, m which one developer tries to assemble two units of land, belonging to different landowners Usually, a developer vlslts each landowner and negotiates over the price to be paid In particular, the developer may negotiate sequentially with each landowner Typically, m such a case the landowner who negotiates last tends to have an advantage over both the first landowner and the developer This 1s because the developer has to commit hlmself to pay a price to the first landowner before he negotiates with the second *The authors have benefited from suggestive comments by Masahlsa FuJita, Miciuhlro Kalyama, Atsuyuki Okabe, Tony E Smith, Konrad Stahl, Xavier Vives, anonymous referees and participants of the first meeting of the Applied Regional Science Conference, wl-nch 1s gratefully acknowledged 01664J462/90/$03
50 0
1990-Elsevler
Science Publishers
B V (North-Holland)
538
Y Asann and A Terakl, Optvnal
negotratton orders and land prrces
Such a negotlatlon process 1s very difficult to analyze Smce we can think of many processes (1 e, many extensive-form games) which are equally plausible, and smce the ‘equlhbnum solutions to these games tend to give rise to different prices, this problem has not been analyzed until recently [Sutton (1986)] But several attempts have been made to give some ‘reasonable’ solutions m such sltuatlons Eckart (1985), for example, analyzed a land assembly problem by modeling It as a two-stage game One developer announces an overall rent to the group of landowners The landowners can accept or rqect this offer, then they can counter-propose a rent profile, which the developer must accept or reJect In the latter case, no development occurs In this model, the landowners’ uncertamty with respect to the developer’s potential profit (or his reservation rent) plays a crucial role Due to this uncertainty, the resulting rent must he between the true reservation rent and landowners’ reservation rent Eckart shows that the coahtlon of landowners set a more moderate price than independently acting landowners, so that the development IS more hkely to be realized Moreover he shows m his model that large landholders tend to set low prices Asaml (1988) considered an alternatlve model m which m developers and n landowners must negotiate over rents To simplify the analysis, he Introduced the notion of admlsslble allocations, and determined the set of players who can acquire a posltlve share of economic profit Aslde from land markets, sequential negotlatlon processes have been extensively studied m game theory literature [as surveyed m Sutton (1986)] Rubmstem (1982), for example, considered a sequential negotiation problem, m which two players negotiate over how to split one cake m the presence of time dlscountmg This problem was extended by Bmmore (1985) to threeplayer/three-cake problem, m which only one pair of players IS allowed to dlvlde a cake ’ Along these lmes, the present paper analyzes the outcome of sequential palrwlse negotiations between a single developer and several landowners In particular, several examples based on the model m Asaml (1988) are analyzed using the concept of a Nash bargatnmg solutton [Nash (1950, 1953), Roth (1979)] 2 Let I = 1,2 be two players negotlatmg how to share a total (pooled) payoff of T Let x, be t’s share (shared payoff) Assume also that d both players fall to reach an agreement (and thus fall to share the total payoff T), then player 1 can obtam a status-quo payoff u, Note that u,‘s are not given to players when they reach an agreement The Nash bargamng solution, x* =(x:,x:), IS then defined by
‘See also Samuelson (1980), Fudenberg Rubmstem (1985), Rubmstem and Wohnsky ‘See also RIddell ( 198 1)
and Txole (1983), Roth (1985) and Hailer (1986)
and
Schoumaker
(1983),
Y Asamz and A Terakl, Optimal negotratlon orders and land prices
539
x* solves the problem
max(x1- 4) (x2-h), x
subject
to
x1 +x2 = T,
Xl 2v,,
x,zv,
This solution concept IS often used to predict negotiation outcomes [cf Harsanyl (1977), Bmmore (1980a, b, 1981)] It 1s important to emphasize here that x*‘s and u,‘s as well as T can be negative m this problem The condltlon to make this solution concept visible 1s only that T~u, + v2 3 To illustrate the nature of results, we begin with a simple example employmg the Nash bargammg solution m section 2 Several results are described m section 3 Given a marginal profit of 1 attamable from development, it 1s shown that d the developer has n alternative plots available, then his payoff (share) approaches 1 as n increases On the other hand, if the developer needs all the n land units, then his payoff approaches 0 as n Increases In some cases, it 1s shown that it 1s optimal for the developer to procure a land umt which 1s never used m the development, so that he can strengthen his bargaining posItIon An alternative model is analyzed m sectlon 4, m which the developer reserves the right to cancel all price agreements Results quahtatlvely similar to the above are obtained m the alternative model It, however, turns out that the order of negotlatlon becomes more Important than m the original model m some cases Moreover, the payoff of the developer IS larger than or equal to that m the original model The present model 1s extended, m sectlon 5, to include the case m which landowners possess a variety of sizes of land umts In particular, it IS shown that the developer ~111 choose to negotiate with the landowner with the largest land umt, if the developer has the choice between one large umt and two small units Several concludmg remarks are stated m sectlon 6 All proofs are relegated to the appendix 2. Model and solution concept Consider a simple land development problem m which one developer seeks to assemble a set of k contiguous land umts or more m order to obtain a fixed economic profit of 1 If the developer falls to assemble a set of k contiguous land units, then no economic profit 1s assumed to be realized, that is, the sum of all players’ payoffs IS zero There are n landowners, and 3For example, suppose that T= -20, u, =0 and II*= - 100 We may Interpret this sltuatlon as if two players agree on the negotlatlon, then their total loss IS only $20, whde otherwise, player 1 ~111 not lose anythmg, but player 2 wdl lose $100 The Nash bargammg solution IS that x, =40 and xz = -60 That IS, player 2 can reduce his loss to $60 by paymg the ‘reward’ of $40 to player 1
540
Y Asaml and A
Terakl,
Optrmal
negotratron
each zth landowner owns the single land unit, Thus the land market consists of n contiguous denoted by player 0, and the lth landowner feasible payoff (share), p E R”+ ‘, must satisfy the
orders and land prrces
[l- 1, I), m the interval [0, n) land units The developer 1s 1s denoted as player I Any condltlon that
Here p, (1~0) can be interpreted as the land price of the zth land unit, and p0 1s the developer’s net development profit The land-development game described above 1s completely specified by k and n, and 1s thus designated as G(k, n) To illustrate the solution concept, consider the simplest case G( 1,1) In this case only one developer and one Iandowner negotiate over how to divide the profit 1 If they fall to agree, then the developer cannot develop the land, and both will get zero profit In other words, the status-quo point of this negotiation 1s assumed to be (0,O) 4 Recalling the definition of the Nash bargaining solution (l), it 1s easily verified that the solution 1s given by (l/2, l/2), 1 e , that they equally divide the profit Since m this case both players seem to have an equal ‘bargaining power’, It 1s reasonable to expect this outcome To illustrate a more complicated problem, consider a game, G(1,2) In this case the developer has a choice as to which land unit to buy Suppose that he decides to negotiate first with landowner 1 This negotlatlon order 1s denoted by [ 1,2] 5 For analytical slmplmty, we will assume that (1) first the developer chooses the order of negotiations, which IS announced to everyone, (11) then the developer negotiates successively with each landowner according to this order, (111) once an agreement 1s reached m each negotiation, the result 1s announced to everyone (including the price agreed upon), (iv) the developer cannot proceed to the next negotiation without settling the present negotlatlon, 1e, he must decide whether or not to buy, and must commit himself to the price level if he buys, and (v) no participant can change the agreement or the rejection after the negotlatlon 1s completed (1 e, no renegotiations are possible) Returning to the example above, suppose that the developer chooses order [l, 21 If he were to negotiate with player 2, then this must imply that he failed to agree m the first negotlatlon This case 1s thus the same as G(l, 1) *The first component represents a value for the developer, whde the rest represent values for landowners This not&on 1s used throughout this paper 51n general [r,.rz, ,rJ means that the developer starts negotlatmg with the r,th landowner, then with the r,th, , and finally with the r.th
Y Asamr and A Terakl, Optlmal negotratlon orders and land prices
541
and both players 0 and 2 can expect l/2 of the profit In the first negotiation, smce the developer can expect the payoff l/2 if this negotiation falls, the status-quo is (l/2,0, *) 6 Recallmg (l), we obtain the solution (3/4, l/4, *) Since the order does not matter for the developer, we can assume that the developer will choose [1,2] or [2, l] with equal probablhty Based on this assumption, we obtain finally the expected payoff E=(3/4, l/8, l/8) The result heavily depends on the assumption on the ‘off-negotlatlon’ results, that 1s negotlatlon results which never occur m the resulting negotiation process Without consldermg such cases, it 1s impossible to derive appropriate statusquos This 1s why we have to consider the negotiation between players 0 and 2 which never occurs m the resulting negotiation process m the example above It 1s mstructlve to consider another example Consider a game, G(2,2) In this case the developer has to assemble two land units Suppose that he decides to negotiate first with landowner 1, 1 e, that the negotiation order 1s [1,2] Assume that the developer agreed with landowner 1 upon the purchase of land with price, p1 The developer moves to the negotiation with landowner 2 If this negotlatlon falls, then the developer cannot obtain the economic profit of 1 It follows that the developer’s payoff amounts to be -pl, and developer 2’s payoff to be 0 In other words, the status-quo IS ( -pl, *,O) On the other hand if the negotiation succeeds, then they can share the profit of 1 Suppose that landowner 2 takes p2 Then 1 -p2 will be left to the developer The total payoff of two players is, hence, given by (1 -p2 -pl) fp, = 1 -pl By substltutmg T, u1 and u2 with l-p,, -pl and 0, respectively, m (I), the Nash bargaining solution IS derived as x1 = 1/2-p, and x2= l/2, that is, (1/2-p,, *, l/2) Turning to the first negotiation between the developer and landowner 1, we may proceed as follows If the negotiation falls, then the developer has to give up the economic profit of 1, and hence the status-quo 1s (O,O, *) If the negotiation succeeds, then landowner 2 can get the price, pl, and the developer will get 1/2-p, after all It follows that the total payoff 1s (1/2-p,) +pl = l/2 Recalling (l), we obtain the solution, (l/4, l/4,*) Combmmg the results above yields the solution, (l/4,1/4,1/2) The expected payoff can be calculated by consldermg all possible negotiation orders
3. Bargaining solutions in the fundamental
model
For some simple cases, it 1s possible to generalize the result described above The first case concerns a sltuatlon m which the developer needs only one ‘In this context, player 2’s value 1s not directly related to the status quo of the two players the current analysis In such a case, ‘t’ 1s used to denote the correspondmg element
m
Y Asamr and A Terakl, Opttmal negotlatron orders and land prxes
542
land umt One expects that as the number of landowners Increases the developer’s payoff approaches the total profit 1 This expectation IS confirmed by the followmg proposltlon Proposition 1 In G(l, n), player O’s share and the first negotiated player’s share are 1 - l/2” and l/2”, respectively lhe expected payoff 1s given by (E, l=O, ,n), where
E, = 1 - l/2”, E,=l/(n2”),
(3) l= 1,
,n
(4)
Next consider a sltuatlon m which In this case, it 1s shown below that order of negotlatlons, the larger 1s developer’s share approaches 0 as the Proposltlon share
2
In G(n,n),
If [l, 2,
PO =
of negotlatlons
payoff
O’s
(5) r=l,
,n
(6)
1s given by (E,
E, = l/2”,
E,=(l-1/2”)/n,
does not affect player
the share, p, 1sgiven by
l/2”,
pl=1/2”+‘-‘, The expected
the order
,n] IS adopted,
the developer needs all the land umts the later a landowner appears m the the share he obtams Moreover, the number of landowners increases
l= 0,
, n), where (7)
l=l,
,n
(8)
The main reason for the different results m Proposltlons 1 and 2 1s that m G( 1, n) land units are substitutes for the developer, while m G(n, n) land units are complements The developer, hence, has a number of alternative choices for the development m G( 1, n) for n 2 2, which 1s not the case m G(n, n) The two cases considered above are very special cases, where the order of negotiations does not change the developer’s share If 2 5 k sn1, however, the order of negotiations can be a very important strategic variable for the developer Consider G(2,3) In this case, the developer must buy landowner 2’s land unit, but he needs only one of the peripheral land units In other words, player 2 has a stronger bargaining position than players 1 or 3 Intmtlvely
Y Asaml and A Terakl, Optrmal negotlatlon orders and land prices
543
the developer appears to be better off by begmnmg the negotlatlon with player 2, because it would appear that landowners m the earher negotlatlons have weaker bargaining posltlons Surpnsmgly, this mtultlon 1s false, as 1s shown m Proposltlon 3 The order of negotlatlons does not affect player O’s share, although player 2’s share 1s actually lower d negotiations are held with him first Proposztzon 3 In G(2,3), the order of negotzatzons does not affect player O’s share If negotzatzons are held wzth player 2 first, then hzs share IS 318, otherwise l/2 The expected payoff IS given by (318, l/12,1 l/24,1/12) In Proposltlon 3, it 1s observed depend on the order of negotlatlons property holds for more complicated to consider the case of G(2,4) As 1s order becomes an important strategic
that the developer’s payoff does not Hence one may ask whether this situations In particular, it 1s of interest shown m the followmg proposltlon, the variable for the developer m this case
Proposztzon 4 In G(2,4), player O’s optzmal negotzatzon orders are all those whtch ezther (I) start from a peripheral landowner (I e , 1 or 4), or (II) start from a central landowner (z e 2 or 3) and proceed to an adjacent landowner In all cases, player O’s share IS l/2, and the expected payoff zs gzven by (l/2,7/80, 13/80,13/80,7/80) One example of a non-optimal negotiation order 1s [3,1,4,2] This 1s mtultlvely understandable since player 0 will waste money by buying land unit 1, which 1s never used m the development With this m mmd it might appear surprlsmg that the order [4,1,3,2] 1s optimal, since m this order, player 0 will also waste money by buying land unit 1 One different aspect m this order, however, 1s that by negotiating with player 1, player 0 succeeds m reducing player 4’s negotiation power In particular, if player 4 does not agree, then the remaining order [l, 3,2] 1s still an optimal negotiation order for the subgame that player 4 1s excluded from the orlgmal game Stated differently, all the remaining land units are still effectively available for the remaining negotiation Hence player 0 has a higher status-quo position for the first negotiation In the former order, however, d player 3 rejects the land procurement, then the remaining order [1,4,2] 1s not optimal for one land unit can never be used for the development even though there are three remaining land units Player 0, hence, has a worse status-quo position m the first negotiation These observations are consistent with the mtultlve prmclple that the negotiation order should be so selected that the remaining negotiation order at each step continues to be advantageous If we fix the order of negotiation as [l, 2, , n], then it 1s possible to calculate the shares for the cases of G(2,n)
Y Asamr and A Terakl, Optimal negotlatlon
544
Proposmon 5 In G(2, n), If player share p”, IS given by
0 adopts
orders and land prrces
the order
[l, 2,
,n],
then the
p;=I-3v,_1-v,_2,
(9)
(11) p:=O,
for
1>3,
(12)
where Uk
=([(l + &)/41k-[(1-$)/41k)i(2&)
4. An alternatwe
(13)
model
So far it has been assumed that the developer must pay all prices agreed upon, even If he falls to assemble a feasible set of land umts for development Smce there IS only one developer m the model, It may be argued that this IS too restrlctlve an assumption If we Interpret a prior commitment to buy land as d way to reserve this land, then a developer does not necessarily have to reserve land if there IS no other competitor To modify this sltuatlon, we adopt the alternatlve assumption that the developer has the right not to pursue the project and cancel all prices (1 e , all contracts to buy) However, it will be assumed that if the developer proceeds with development, then he must pay prices not only to the landowners of relevant land units, but also to all other landowners with whom he has agreed to buy land units ’ Based on this ‘condltlonal contract’ assumption, we must modify all the results For a notational convenience, this alternative game ~111 be designated as G’(k, n) Proposltlon share are (E, I =O,
6
In G’(1, n), player
and , n), where
1 -l/2”
l/2”,
O’s share
respecmely
and the first The
expected
negotiated payoff
player’s
IS given
by
‘We can consider stdl another scheme m winch the developer has a right to cancel buying the land for any subset of land units However m such a case, we may face some mdetermmacy problems Consider for example G( 1.2) If the developer selects the order [1,2] and lf the price of land umt 1 IS settled as X, then the price negotlatlon with landowner 2 wdl result m d price of u/2 Now the questlon IS how to determme the value of x The status-quo IS (l/2,0,*), and money left for negotlatlon 1s 1 But then IS Y equal to l/27 It seems that the Nash bargammg solution concept Itself IS Irrelevant here, for the result of the second negotlatlon cannot affect the solution Hence, to deal with such a case, it appears that we must take a non-cooperative extensive-form game approdch
Y Asamr and A Terakz, Optimal negotlatlon orders and land prrces
E, = 1 - l/2”, E,= l/(n2”),
545
(14) I= 1,
,n
(15)
It 1s worth noting that the result m Proposltlon 6 1s exactly the same as m Proposltlon 1 In other words, if k= 1, then shares are not affected by whether the developer has a right to cancel buymg commitments The reason 1s that the developer’s status-quo position never becomes negative even m G( 1, n) If kz 2, however, this modlficatlon of the assumptions crltlcally affects the result Proposztzon 7 share If [1,2,
In G’(n,n), the order of negotzatzons does not affect player O’s , n] zs adopted, the share, p, IS gzven by
p,=l/(n+l),
z=O,l,
,n
The expected payoff IS gzven by (E, z=O, E,=l/(n+l),
z=O,l,
,n
(16) , n), where (17)
It IS somewhat surprlsmg that the order does not affect any player’s share This 1s due to the fact that the developer does not have to make a commitment to buy By this assumption, the status-quo position for each player 1s zero m any negotiation, and hence the profit 1s shared equally In G(2,3), it was observed that the order of negotiations does not affect player O’s share The followmg proposltlon shows that this 1s no longer valid m G’(2,3) Proposztzon 8 In G/(2,3), the optzmal order of negotzatzons for player 0 IS to start the negotzatzon wzth player 1 or 3 Then players 0 and 2’s shares are the same and equal 419 The first negotzated landowner’s share IS l/9 The expected payoff IS gzven by (419, l/18,4/9,1/18) It seems quite uruntultlve that the negotiation order starting with player 2 IS not optimal In this game player 0 and player 2 get the same share m all cases, for the status-quo 1s zero m any case for both players (which 1s the mam difference from the original model) Then the optimal order must involve some orders which weaken player l’s (or 3’s) negotiation power This 1s the reason why the developer should start negotiation with player 1 (or 3) One may wonder why both [l, 2,3] and [l, 3,2] are both optimal m G’(2,3) Suppose first that the first negotiation falls Then It 1s evident that both orders are optimal m the remaining negotiation Now suppose that the developer decides to buy land unit 1 In this case, whether the development
546
Y Asamr and A Terakl, Optrmal negotlatlon
orders and land prices
project can be carried out or not depends totally on the negotiation with player 2 In other words, buying land unit 3 does not help the developer at all Since the developer can freely decline to buy land m any negotiation (or he can mslst on buying the land with price zero), [ 1,2,3] and [l, 3,2] are both equivalent to [l, 23 provided that the developer buys land unit 1 with a price under [l, 2,3] Note that m G/(2,3) the developer 1s better off by first negotiating with player 1 or 3 Hence one may ask whether this property holds for more complicated cases To allow a meaningful comparison, let us consider the game G’(2,4) Proposltzon 9 In G’(2,4), player O’s optimal negotlatlon orders are all those starting from a peripheral landowner (1 e, 1 or 4) followed by an adjacent landowner (1 e, 2 or 3) In all cases player O’s share 1s 26145, and the expected payoff 1s given by (26145, l/l 5,13/90,13/90,1/l
5)
The optimal orders are [1,2,3,4], [1,2,4,3], [4,3,2, l] and [4,3,1,2] These orders satisfy the condltlons that (1) even if the first negotiation falls, the rest of the order 1s optimal m the remaining subgame, and (11) the developer does not have to pay a price for any unnecessary land unit Hence the basic principle suggested by the model m section 3 above 1s still valid One common property observed m both models 1s that the optimal order of negotiation should be so determined that if the first negotiation falls, then the remaining order continues to be optimal m the remaining subgame If we employ this prmclple recursively, then it follows that for any negotiation which falls, the remammg order continues to be optimal This prmclple, which can be called ‘maximum principle of negotiations’ (or a ‘subgame optimal prmclple’), has an mtultlve appeal For if the developer follows this prmclple, then he can maintain the strongest status-quo position m any negotiation To state this principle more precisely, let O(m) be the set of all possible permutations of [l, ,m], which denotes all the possible negotiation orders Define the set of optimal orders, o*(m), such that for any where n(0) 1s the developer’s share c7E O*(m) c O(m), n(a) =mdxoE 8(mj n(O), associated with the order 0 Take CJEO(m) and z E O(m + 1) If CJand 7 satisfy ,x,,,+J 1s the same as [a,, that [z,, ,oml or Co,+ 1, a,+ 11, then ~7 IS said to be embedded in T Define the set of orders, @‘(mil), m O(m+ 1) m which some optimal order m O*(m) 1s embedded O+(m+
1) = (0~ O(m + 1) there exists (TE O*(m) embedded
With this notation,
the prmclple
such that B 1s
m 0) above
1s convemently
expressed
as
Y Asamr and A Terala, Optimal nqotratron orders and land pees
541
Maximum Prrnclple of Negotlatron max ZZ(O)= max Z7(cr) aEf3+(a-l) BE6W Though It IS not proved here,’ the order [l, 2, ,n] seems to be optlmal for G’(2,n), for this order sat&es the principle above If we fix the order of , n] (which 1s an optimal order for n= 2, 3 and 4), negotiations as [ 1,2, then It 1s possible to calculate the shares for the case of G’(2,n) Proposltlon 10 In G’(2,n), If player 0 adopts the order [1,2, share p” IS gzven by
,n], then the
P”o= Yll?
(18)
P;=(1-2Y”-,+V”-J/(3+Y,-J,
(19)
P);‘(l
(20)
+Y,-,)(~-Y”-,)/(3+Y”-,)~
pf=O
for
(21)
123,
where y, IS recursively defined by y,=(l
+y,-I)(1
+y,-,)/(3+y,-2)
(22)
and (23)
Yo=Y,=O Moreover these terms satisfy lim p;= Il’a)
1,
(24)
lim py = lim p; =O, n-CC n-m
(25)
llm (P;/P;)=($“-CC
I)/4
(26)
The result that the developer’s share approaches 1 as n Increases IS quite understandable For n large, even if the developer falls m the first negotlatlon, he still has a lot of other possible assemblies of land umts, which strengthens his bargaining position accordingly It 1s of interest to note that ‘At a first glance, It may not appear a dlffkult problem argument The proof, however, will mvolve a combmatorlal
to prove by a mathematical mductlon analysis, which IS hard to conduct
548
Y Asam and A Terakl, Optrmal negotlatlon orders and land pruzes
h-n, + (P;/P;)=(&-
I)/4 Th IS means that when n-+ co, player 2’s land umt 1s worth (1 + a) times that of player 1 (even though the absolute value of prices approaches zero) Since player 2’s land unit 1s necessary m order to develop player l’s land unit, it 1s easily seen that player 2 should obtain a larger share than player 1 The proposition says further that even if n goes to mfimty the ratio between these prices never goes to zero 9 m
5. Extensions
to various land unit sizes
So far, it has been assumed that all landowners possess land units of the same size In this section, this assumption 1s relaxed to include cases m which landowners own various sizes of land units Toward this extension, several modliicatlons m notation will be introduced m the followmg Let a be the size of land mmlmally necessary to yield a fixed profit 1 to the developer (called player 0) It 1s assumed as before that the developer can gam only one unit of profit by developing a size of land larger than or equal to a There exist n landowners Each landowner, 1, possesses a land unit of size S, All land units he on a line In particular, landowner I owns all land m the interval, CC; = 1 S, - S,, It, = 1 S,), m the total land of [O,cJ= 1 S,) Since this extended version of land development game can be characterized by the necessary land size, a, and all landowners’ land unit sizes, S=(S, J = 1, I 4, the model 1s designated ds g(u,S) hereafter One might argue that if a m the landowner possesses a larger unit, then he has an advantage negotiation of land price, when the size of land unit varies among landowners To examme this mtultlon, we begin with an analysis using assumptions similar to that m section 3 Suppose again that the landowner cannot cancel the negotiated prices The other assumptions on the negotiation process are the same as m section 3 To simplify the argument, let us introduce the following notation Let L(~,J) be the set of landowners ordered from I to J, where 1_I J L(t,J)={t,
,J1-
(27)
Then the family, L, of mmlmal developer can succeed to develop follows
L=
L(L,J)
91t can be readdy
i
k=,
S,Za,
i k=t
sets of landowners, land to obtain profit
Sk-S,
i
with whom the land 1, can be expressed as
Sk-S,
k=r
shown that tlus hrmt 1s the same for the basx
3
model above
Y Asamzand A Terakl, Optimal negotlatton orders and land prrces With this notation, it 1s shown that if n=3, and rf a landowner included m any mmlmal set m the family, L, then his land price irrespective of the order of negotiation Proposztzon 11 In g(a,S) wzth S=(S, zn L, then J’S share, p,, zs zero
I= 1,2,3),
If
J IS
549
IS not 1s zero
not included zn any L
Proposltlon 11 implies that d a landowner’s land unit 1s not necessary for the development, then he is practically ignored m the negotiation The Shapley value [Shapley (1951, 1953)‘J 1s often used to find the relative negotiation powers among players It 1s seen that the share m our negotlatlon game, and the Shapley value exhibit a slmllar feature In particular, a large landowner has an advantage m the negotiation only if the required land size, a, 1s relatively small Proposztzon 12
In g(a, s) wzth S= (S, l= 1,2,3)
If asmax, S,, then the landowner who owns the largest land unzt obtazns the largest Shapley value among all landowners (II) If a>max,S,, then landowner 2’s Shapley value zs the largest among all landowners (I)
The proposltlon above means that if the required land size, a, 1s smaller than the largest land unit, then the share of the largest landowner 1s the largest of all landowners, and if the required land size 1s larger than the size of the largest land unit, then the developer has to negotiate with several landowners and thus the middle landowner receives the largest share As m the alternative model m section 4, we can also consider an alternative extended model with a variety of land sizes, m which the developer preserves a right to cancel all the negotiated results at the end We denote this extended model by g’(a,S) Then we can analyze optimal orders of negotiation as follows Proposztzon 23
In g’(a,S) wzth S=(S,
z= 1,2,3)
Zf max(S,,S,)
In the alternative
extended
model, g’(a,S),
if S=(S,,S,,S3)
satisfies
one of
the condltlons m Proposltlon 13, then there exist (m) above reduces to G’(2,3), for the developer combmatlons of land units, il, 2) or (2,3} On developer can choo,se between one larger un:t and and (ii) above], then it is ddvantageous for him to landowner first
optimal orders The case can choose one of two the other hand, d the two rmall unit3 [l e, (1) negotiate with one large
6. Concludmg remarks
This paper has analyzed a simple class of sequential negotiation games, m which one developer negotiates land prices with landowners m order to develop a project worth one unit of profit It was shown m Proposltlon 1, that if the developer has n alternative plots avdilable, then his share approaches 1 as II increases The ‘converse’ case was tredted m Proposition 2 If the developer needs all the n land units, then his share approaches 0 as n increases It was shown that the order of negotlatlons may or may not be a strategic variable even If the developer has some alternatives of development patterns each of which consists of several land units In particular, m G(2,3) the order does not affect player O’s share (m Proposltlon 3), while m G(2,4) it does (in Proposltlon 4) The basic prmclple seems to be that player 0 should select the order so that he can retam an optimal order after the first negotiation An alternative model was analyzed rn section 4 m which the developer retains a right to cancel all prices It was shown m Proposltlon 6 that if the developer needs only one plot, then the result remains the same as m the former model If the developer needs all the plots, however, then the players equally divide the potential profit (m Proposltlon 7) In G’(2.3) and G’(2,4), the order of negotlatlons become a crItica strategic variable for the developer In both cases, it was observed that an order which satisfies the ‘maxlmum prmclple of negotlatlon’ if optimal This principle, namely that ‘the order of negotlatlons should be selected so that even if any negotiation falls, the remammg order 1s still optimal m the remammg subgame’, appears to be a fundamental rule for negotiation processes In section 5, an extension of the basic model 1s analyzed m which the sizes of land units are allowed to vary It was shown in Proposition 11 that the price of ‘redundant’ land units which does not influence the posslblhty of development is dlwdys zero Moreover, it is optimal for the developer to start the negotiation with the landowner with the largest land umt provided that the developer hds the choice between one large unit and two small units (Proposition 13) A very important class of models are left for future research, namely models which allow for collusion of players and/or renegotiation In our models, for example, if the developer and landowner collude secretly, then it
Y Asamr and A Terakr
Optlm&l negotlatlon
orders and land prwes
551
may benefit both players If this 1s the case, then some players may suspect the existence of a secret collusion Such a model with Imperfect information 1s a new lme of research to be conducted m the future Even wlthout the posslblhty of any secret collusions, we may derive a different result by taking the posslblhty of open collurlons mto account On the other hand, If we permit renegotlntlon, then we may derive a more even dlstrlbutlon of share Ii)
Appendix: Proofs of propositions A I
Proof
of Proposltlon
1
The argument 1s by mathematical mductlon For n= 1. the result follows from section 2 Moreover, if the assertlon holds for all ni k, then, for G( 1, k + l), the status-quo point of the first negotlatlon 1s (1 - 1/2k. 0, *. *), where the negotlatlon order 1s [1,2, ,n] Hence it follows that pi = 1 - 1/2kf1 and p1 = l/2 k+’ Fmally noticing that the order does not matter for the developer, the result follows’ QED A 2
Proof of Proposltlon
2
Again, employing mathematical mductlon, observe that for n= 1 and 2, the result follows from yectlon 2 Moreover, If the assertion holds for all n < k. then m G(k + 1. k+ 1) the status-quo point of the first negotiation 1s , *), where the negotiation order 1s [1,2, ,n] Since other players (0,0, *, (than players 0 and 1) will get the total share of 1- lj2k, only 1/2k 1s left for both to share Hence it follows that p0 = 1/2k+1 and p1 = 1/2k+1 Finally noticing that the order does not matter for player 0, the result follows QED A 7 Proof of Proposltlon
3
If the order is [2,1,3], then the share is (3/8, l/4,3/8,0) If the order IS [1,2,3], then the share 1s (3/8, l/8,1/2,0) If the order 1s [1,3,2], then the share 1s (3/S, 1/8,1,‘2,0) All other cases can be verified similarly QED A 4
Proof of Propontlon
4
If the order is [1,2,3,4], then the share 1s (l/2, l/S, 3/8,0,0) If the order 1s Cl, 3,2,4], then the share 1s (l/2, l/8,1/4,1/8,0) If the order 1s [2,1,3,4-J, then the share 1s (l/2, l/4,1/4,0,0) If the order 1s [2,4,1,3], then the share 1s (7/l 6, l/4,3/16,0, l/8) All other cases can be verified similarly QED
‘?ke Asaml
(1988) for a study along thts hne
Y Asaml and A Terakz, Optimal negottatlon orders and land prrces
552
A 5
Proof of PropoAltlon
5
For n = 2,3, the py’s are given above For larger relationship among p& py and p; for each n P”o’ l
Solvmg
A 6
values
of n, we have the
=(l +p”o-‘+2p1;)/4,
this system with the mltlal
Proof of Proposctlon
condltlons,
we have the result
QED
6
For n= 1, the status-quo 1s (0,O) and thus pO=pl = l/2 assertion holds for all ns k, then m G’( 1, k + 1) at the first status-quo IS (1 - 1/2k, 0, *, , *) given the negotiation order it follows that pO= 1 - 1/2k+1 and p1 = 1/2k+’ Since the matter for the developer, the result thus follows by induction
A 7 Proof of Proposltlon
Moreover, if the negotiation, the [l, 2, ,nl, and order does not
QED
7
For n= 1, p, = l/2 for I =O, 1 If the assertion holds for n 5 k- 1, then m *) given negotiaG’( 1, k) at the first negotiation, the status-quo 1s (0,0,*, , k] Moreover, if r IS the price of lani unit 1, then all tion order [l, 2, players other than players 0 and 1 will get a total share of (k- 1) (1 -r)/ (k+ 1) Hence only r +( 1 - k)/k 1s left for players 0 and 1 to share, and it follows that r =(r +( 1 - r)/k)/2 or r = l/(k + l), which implies that p, = l/(k + 1) , k Again noticing that the order does not matter for player 0, the for 1=0, result follows by mductlon QED
A 8
Proof of Proposltlon
8
If the order 1s [2,1,3], then the share 1s (3/7, l/7,3/7,0) If the order 1s [l, 2,3], then the share is (4/9, l/9,4/9,0) If the order 1s [l, 3,2], then the share IS (4/9, l/9,4/9,0) Other cases can be calculated similarly, and the result follows QED
A 9
Proof of Proposltlon
If the order
9
1s [ 1,2,3,4],
then
the share
1s (26/45,2/15,13/4&O,
0) If the
Y Asaml and A Terakl, Optrmal negotratlon orders and land prwes
order 1s order IS order 1s [2,4,1,3], slmllarly
A 10
553
[1,3,2,4], then the share 1s (90/161,3/23,30/161,20/161,0) If the [1,4,2,3], then the share 1s (13/23,25/207,13/69,0,26/207) If the [2,1,3,4-J, then the share IS (4/7,4/21,5/21,0,0) If the order 1s then the share is (12/23,4/23,13/69,0,8/69) All other cases can be verified QED
Proof of Proposmon
10
For n=2, 3 and 4, the proposltlon follows from Propositions 7, 8 and 9 Moreover, if the assertion holds for all n 2 k- 1 (2 4), then m G’(2, k) if the first negotiation falls, the remaining case reduces to G’(2, k- 1) Thus the status-quo of player 0 1s y,_ 1 Suppose the first negotiation results m a price of x If the second negotiation falls, then the remaining case reduces to G’(2, k-2) with a total payoff of (1 -x) Hence it follows that the status-quo of player 0 m the second negotlatlon 1s given by (1 -x)y,-,, so that the total share left for players 0 and 2 1s 1 -x Hence players 0 and 2’s shares are (1 -x) (1 + y, ~ J2 and (1 -x) (1 - y, _ J2, respectively In the first negotlatlon, the status-quo of player 0 1s y,_ 1 as shown above The total share which can be shared by players 0 and 1 1s 1 - (1 -x) (1 -y,_ J/2 It follows that x=(1-2yk_,+yk_,)/(3+y,_,) Hence
If O 1s monotone decreasing, and that {y,} 1s bounded and monotone increasing, and hence convergent Let Y be the hmlt of this sequence where 0 5 Y 5 1 Y must satisfy Y = (1 + Y) (1 + Y)/(3 + Y) and hence Y=l, and hm,,,z,=O Define r,=z,/z,-, If 1/2-~r~_~
have It
we
[r._1(4-z,_,)]
+J5)+(,/5-3)z,_J4]/ Ir,-(1+,/‘$/4~=hm,,,
(r,_,-
+Js) hm,+, r,-J
(1+$)/4I/C(l
lmplymg (2 -z,
r--(1 +J5)/4=[1-4r,_,/(l follows that hm,,,
that
554
A I1
Y Asamr and A
Terukr,
Proof of Proposltlon
Optrmal
negotratlon
order,
and land prrc~\
11
(a) If S, 5 S, 5 S, holds, then If UPS,, then everyone IS Included m some L m L, If S,
(b) If S, 5 S, 5 S, holds, then (1) if ulS,, then everyone IS included m some L m L, (II) if S,
A 12
cases
can
be
Proof of Proposltlon
12
(a) If S, 5 S, 5 S, holds,
(I) if uzS,, then (11) if S,
verified
similarly
Thus
the
proposltlon
1s
QED
then
the Shapley value 1s given by (3/4, l/12, l/12, l/12), then the Shapley value IS given by (2/3,0,1/6, l/6), and UPS, +S,, then the Shapley value 1s given by (7/12, and
S, +S,
then
the Shapley
the Shapley value
value
IS given
I$ given
by (l/2,
by (5j12, l/12, l/12,
then the Shapley value IS given by S,+S,
Y Asaml and A Terakl, Optrmal negotratron orders and land prvzes
(b)
555
If S, 5 S, 5 S2 holds, then
If d (111)if (14 If
II) (4 (VI)
asS,, then the Shapley value 1s given by (3/4, l/12,1/12, l/12), S, SusS,, then the Shapley value IS given by (2/3,0,1/6, l/6), S, CUSS,, then the Shapley value 1s given by (l/2,0, l/2,0), S,
l/l 2), If S,
IS given
by
by (l/4, l/4,
l/4,1/4) (c) All other cases can be verified result QED A 13
Proof ofPropos&on
similarly,
and hence
we have the desn-ed
13
The proposltlon can be verified by exammmg the proof of ProposItIon 11 QED
all cases exhaustively
as m
References Asaml, Y, 1987, Game-theoretic approaches to bid rent, Ph D dlssertdtlon (Department of Reglonal Science, Umverslty of Pennsylvdma, Phlladelphla, PA) Asaml, Y, 1988, A game-theoretic approdch to the dlvlslon of profits from economic land development, Regiondl Science dnd Urbdn Economics 18, 233-246 Bmmore, KG, 1980a, Nash bargaining theory I, ICERD dIscussIon paper. no 9 (London School of Economics, London) Bmmore, KG , 1980b, Nash bdrgammg theory II, ICERD dlscuqslon paper, no 14 (London School of Economics, London) Bmmore, KG, 1981, Nash bdrgdmmg theory 111, ICERD dIscussIon paper, no 15 (London School of Economics, London) Bmmore, K G, 1985, Bargdmmg and coahtlons, in A E Roth, ed , Game-theoretic models of bargammg (Cambndge Umverslty Press, Cambridge) 267-304 Eckdrt, W, 1985, On the land assembly problem, Journal of Urban Economics 18, 364378 Fudenberg, D and J Tlrole, 1983, Sequential bargaining with incomplete mformation, Review of Economic Studies 50,221-247 Hdller, H , 1986, Non-cooperdtlve bargaming of N 2 3 pldyers, Economtcs Letters 22, 11-13 Hdrsanyl, J C, 1977, Ratlonal behavior and bdrgammg equthbrmm in games dnd SOCldl situations (CambrIdge University Press, CambrIdge) Nash, J F, 1950, The bargammg problem, EconometrIca 18, 155-162 Nash, J F, 1953, Two-person cooperative games, Econometrica 21, 128-140 RIddell, WC, 1981, Bargammg under uncertainty, American Economic Review 71, 579-590 Roth, A E , 1979, Axlomatlc models of bdrgdmmg (Springer-Verlag. Berlin) Roth, A E and F Schoumaker, 1983, Expectations and reputation5 m bargammg An experlmental study, American Economic Review 73, 362-372 Rubmstem, A, 1982, Perfect equthbrtum in d bargammg model, Econometrtca 50, 97-110 Rubmstem, A, 1985, A bargammg model with mcomplete mformatlon about time preferences, EconometrIca 53, 1151-1172 Rubmstem, A and A Wolmsky, 1985, Equlhbrlum m a market with sequential bargamrng, Econometrica 53, 1133-l 150
556
Y Asamc and A Terakr, Optrmal negotratmn orders and land przces
Samuelson, P A, 1980, First offer bargams, Management Science 26, 155-164 Shapley, L S, 1951, Notes on the n-person game, II The value of an n-person game, RAND pubhcatlon RM-670 (RAND Company, Santa Momca, CA) Shapley, LS, 1953, A value for n-person games, m H Kuhn and A W Tucker, eds, Contrlbutlons to the theory of games, Vol 2 (Princeton University Press, Princeton, NJ) 307-317 Sutton, J (1986), Non-cooperative bargammg theory An mtroductlon, Review of Economic Studies 53, 709-724
Science
and Urban
Economics
ON SEQUENTIAL
20 (1990) 537-556
NEGOTIATION
Optimal Negotiation Yasushl
North-Holland
PROCEDURES
Orders and Land Prices* ASAMI
Unrterslty of Tokyo, Tokyo, Japan
Aklhlro Burldmg Research Institute, Received
TERAKI
Mzmstry of Constructmn, Tsukuha, Japan
May 1988, Iinal verS,on received Aprd
1990
Models of land procurement by a developer are dnaiyzed The developer IS assumed to require at least a certam Size of contiguous land to obtam a prolit, wtuch he tries to procure by sequential negotlatlon procedures The results suggest a ‘maxImum principle of negotiation’, namely that an optlmal order of negotiation should always satisfy the condition that regardless of whether or not any given negotlatlon fads, the remammg order contmues to be optimal for the rest of negotiations It may be optimal for the developer to procure land unit wluch IS never used m the development, so that he can strengthen tus bargaining position If the Size of landowners’ land units are different and all units are required, the developer will choose to negotiate the owner with the largest land unit to maxlmlze his payoff
1. Introduction The properties of price or rent profiles m markets with finitely many participants have recently analyzed by several researchers [cf Eckart (1985), Asaml (1987,1988)] Examples of such markets include situations where a few large developers are Involved m the procurement of land Consider a simple problem, for example, m which one developer tries to assemble two units of land, belonging to different landowners Usually, a developer vlslts each landowner and negotiates over the price to be paid In particular, the developer may negotiate sequentially with each landowner Typically, m such a case the landowner who negotiates last tends to have an advantage over both the first landowner and the developer This 1s because the developer has to commit hlmself to pay a price to the first landowner before he negotiates with the second *The authors have benefited from suggestive comments by Masahlsa FuJita, Miciuhlro Kalyama, Atsuyuki Okabe, Tony E Smith, Konrad Stahl, Xavier Vives, anonymous referees and participants of the first meeting of the Applied Regional Science Conference, wl-nch 1s gratefully acknowledged 01664J462/90/$03
50 0
1990-Elsevler
Science Publishers
B V (North-Holland)
538
Y Asann and A Terakl, Optvnal
negotratton orders and land prrces
Such a negotlatlon process 1s very difficult to analyze Smce we can think of many processes (1 e, many extensive-form games) which are equally plausible, and smce the ‘equlhbnum solutions to these games tend to give rise to different prices, this problem has not been analyzed until recently [Sutton (1986)] But several attempts have been made to give some ‘reasonable’ solutions m such sltuatlons Eckart (1985), for example, analyzed a land assembly problem by modeling It as a two-stage game One developer announces an overall rent to the group of landowners The landowners can accept or rqect this offer, then they can counter-propose a rent profile, which the developer must accept or reJect In the latter case, no development occurs In this model, the landowners’ uncertamty with respect to the developer’s potential profit (or his reservation rent) plays a crucial role Due to this uncertainty, the resulting rent must he between the true reservation rent and landowners’ reservation rent Eckart shows that the coahtlon of landowners set a more moderate price than independently acting landowners, so that the development IS more hkely to be realized Moreover he shows m his model that large landholders tend to set low prices Asaml (1988) considered an alternatlve model m which m developers and n landowners must negotiate over rents To simplify the analysis, he Introduced the notion of admlsslble allocations, and determined the set of players who can acquire a posltlve share of economic profit Aslde from land markets, sequential negotlatlon processes have been extensively studied m game theory literature [as surveyed m Sutton (1986)] Rubmstem (1982), for example, considered a sequential negotiation problem, m which two players negotiate over how to split one cake m the presence of time dlscountmg This problem was extended by Bmmore (1985) to threeplayer/three-cake problem, m which only one pair of players IS allowed to dlvlde a cake ’ Along these lmes, the present paper analyzes the outcome of sequential palrwlse negotiations between a single developer and several landowners In particular, several examples based on the model m Asaml (1988) are analyzed using the concept of a Nash bargatnmg solutton [Nash (1950, 1953), Roth (1979)] 2 Let I = 1,2 be two players negotlatmg how to share a total (pooled) payoff of T Let x, be t’s share (shared payoff) Assume also that d both players fall to reach an agreement (and thus fall to share the total payoff T), then player 1 can obtam a status-quo payoff u, Note that u,‘s are not given to players when they reach an agreement The Nash bargamng solution, x* =(x:,x:), IS then defined by
‘See also Samuelson (1980), Fudenberg Rubmstem (1985), Rubmstem and Wohnsky ‘See also RIddell ( 198 1)
and Txole (1983), Roth (1985) and Hailer (1986)
and
Schoumaker
(1983),
Y Asamz and A Terakl, Optimal negotratlon orders and land prices
539
x* solves the problem
max(x1- 4) (x2-h), x
subject
to
x1 +x2 = T,
Xl 2v,,
x,zv,
This solution concept IS often used to predict negotiation outcomes [cf Harsanyl (1977), Bmmore (1980a, b, 1981)] It 1s important to emphasize here that x*‘s and u,‘s as well as T can be negative m this problem The condltlon to make this solution concept visible 1s only that T~u, + v2 3 To illustrate the nature of results, we begin with a simple example employmg the Nash bargammg solution m section 2 Several results are described m section 3 Given a marginal profit of 1 attamable from development, it 1s shown that d the developer has n alternative plots available, then his payoff (share) approaches 1 as n increases On the other hand, if the developer needs all the n land units, then his payoff approaches 0 as n Increases In some cases, it 1s shown that it 1s optimal for the developer to procure a land umt which 1s never used m the development, so that he can strengthen his bargaining posItIon An alternative model is analyzed m sectlon 4, m which the developer reserves the right to cancel all price agreements Results quahtatlvely similar to the above are obtained m the alternative model It, however, turns out that the order of negotlatlon becomes more Important than m the original model m some cases Moreover, the payoff of the developer IS larger than or equal to that m the original model The present model 1s extended, m sectlon 5, to include the case m which landowners possess a variety of sizes of land umts In particular, it IS shown that the developer ~111 choose to negotiate with the landowner with the largest land umt, if the developer has the choice between one large umt and two small units Several concludmg remarks are stated m sectlon 6 All proofs are relegated to the appendix 2. Model and solution concept Consider a simple land development problem m which one developer seeks to assemble a set of k contiguous land umts or more m order to obtain a fixed economic profit of 1 If the developer falls to assemble a set of k contiguous land units, then no economic profit 1s assumed to be realized, that is, the sum of all players’ payoffs IS zero There are n landowners, and 3For example, suppose that T= -20, u, =0 and II*= - 100 We may Interpret this sltuatlon as if two players agree on the negotlatlon, then their total loss IS only $20, whde otherwise, player 1 ~111 not lose anythmg, but player 2 wdl lose $100 The Nash bargammg solution IS that x, =40 and xz = -60 That IS, player 2 can reduce his loss to $60 by paymg the ‘reward’ of $40 to player 1
540
Y Asaml and A
Terakl,
Optrmal
negotratron
each zth landowner owns the single land unit, Thus the land market consists of n contiguous denoted by player 0, and the lth landowner feasible payoff (share), p E R”+ ‘, must satisfy the
orders and land prrces
[l- 1, I), m the interval [0, n) land units The developer 1s 1s denoted as player I Any condltlon that
Here p, (1~0) can be interpreted as the land price of the zth land unit, and p0 1s the developer’s net development profit The land-development game described above 1s completely specified by k and n, and 1s thus designated as G(k, n) To illustrate the solution concept, consider the simplest case G( 1,1) In this case only one developer and one Iandowner negotiate over how to divide the profit 1 If they fall to agree, then the developer cannot develop the land, and both will get zero profit In other words, the status-quo point of this negotiation 1s assumed to be (0,O) 4 Recalling the definition of the Nash bargaining solution (l), it 1s easily verified that the solution 1s given by (l/2, l/2), 1 e , that they equally divide the profit Since m this case both players seem to have an equal ‘bargaining power’, It 1s reasonable to expect this outcome To illustrate a more complicated problem, consider a game, G(1,2) In this case the developer has a choice as to which land unit to buy Suppose that he decides to negotiate first with landowner 1 This negotlatlon order 1s denoted by [ 1,2] 5 For analytical slmplmty, we will assume that (1) first the developer chooses the order of negotiations, which IS announced to everyone, (11) then the developer negotiates successively with each landowner according to this order, (111) once an agreement 1s reached m each negotiation, the result 1s announced to everyone (including the price agreed upon), (iv) the developer cannot proceed to the next negotiation without settling the present negotlatlon, 1e, he must decide whether or not to buy, and must commit himself to the price level if he buys, and (v) no participant can change the agreement or the rejection after the negotlatlon 1s completed (1 e, no renegotiations are possible) Returning to the example above, suppose that the developer chooses order [l, 21 If he were to negotiate with player 2, then this must imply that he failed to agree m the first negotlatlon This case 1s thus the same as G(l, 1) *The first component represents a value for the developer, whde the rest represent values for landowners This not&on 1s used throughout this paper 51n general [r,.rz, ,rJ means that the developer starts negotlatmg with the r,th landowner, then with the r,th, , and finally with the r.th
Y Asamr and A Terakl, Optlmal negotratlon orders and land prices
541
and both players 0 and 2 can expect l/2 of the profit In the first negotiation, smce the developer can expect the payoff l/2 if this negotiation falls, the status-quo is (l/2,0, *) 6 Recallmg (l), we obtain the solution (3/4, l/4, *) Since the order does not matter for the developer, we can assume that the developer will choose [1,2] or [2, l] with equal probablhty Based on this assumption, we obtain finally the expected payoff E=(3/4, l/8, l/8) The result heavily depends on the assumption on the ‘off-negotlatlon’ results, that 1s negotlatlon results which never occur m the resulting negotiation process Without consldermg such cases, it 1s impossible to derive appropriate statusquos This 1s why we have to consider the negotiation between players 0 and 2 which never occurs m the resulting negotiation process m the example above It 1s mstructlve to consider another example Consider a game, G(2,2) In this case the developer has to assemble two land units Suppose that he decides to negotiate first with landowner 1, 1 e, that the negotiation order 1s [1,2] Assume that the developer agreed with landowner 1 upon the purchase of land with price, p1 The developer moves to the negotiation with landowner 2 If this negotlatlon falls, then the developer cannot obtain the economic profit of 1 It follows that the developer’s payoff amounts to be -pl, and developer 2’s payoff to be 0 In other words, the status-quo IS ( -pl, *,O) On the other hand if the negotiation succeeds, then they can share the profit of 1 Suppose that landowner 2 takes p2 Then 1 -p2 will be left to the developer The total payoff of two players is, hence, given by (1 -p2 -pl) fp, = 1 -pl By substltutmg T, u1 and u2 with l-p,, -pl and 0, respectively, m (I), the Nash bargaining solution IS derived as x1 = 1/2-p, and x2= l/2, that is, (1/2-p,, *, l/2) Turning to the first negotiation between the developer and landowner 1, we may proceed as follows If the negotiation falls, then the developer has to give up the economic profit of 1, and hence the status-quo 1s (O,O, *) If the negotiation succeeds, then landowner 2 can get the price, pl, and the developer will get 1/2-p, after all It follows that the total payoff 1s (1/2-p,) +pl = l/2 Recalling (l), we obtain the solution, (l/4, l/4,*) Combmmg the results above yields the solution, (l/4,1/4,1/2) The expected payoff can be calculated by consldermg all possible negotiation orders
3. Bargaining solutions in the fundamental
model
For some simple cases, it 1s possible to generalize the result described above The first case concerns a sltuatlon m which the developer needs only one ‘In this context, player 2’s value 1s not directly related to the status quo of the two players the current analysis In such a case, ‘t’ 1s used to denote the correspondmg element
m
Y Asamr and A Terakl, Opttmal negotlatron orders and land prxes
542
land umt One expects that as the number of landowners Increases the developer’s payoff approaches the total profit 1 This expectation IS confirmed by the followmg proposltlon Proposition 1 In G(l, n), player O’s share and the first negotiated player’s share are 1 - l/2” and l/2”, respectively lhe expected payoff 1s given by (E, l=O, ,n), where
E, = 1 - l/2”, E,=l/(n2”),
(3) l= 1,
,n
(4)
Next consider a sltuatlon m which In this case, it 1s shown below that order of negotlatlons, the larger 1s developer’s share approaches 0 as the Proposltlon share
2
In G(n,n),
If [l, 2,
PO =
of negotlatlons
payoff
O’s
(5) r=l,
,n
(6)
1s given by (E,
E, = l/2”,
E,=(l-1/2”)/n,
does not affect player
the share, p, 1sgiven by
l/2”,
pl=1/2”+‘-‘, The expected
the order
,n] IS adopted,
the developer needs all the land umts the later a landowner appears m the the share he obtams Moreover, the number of landowners increases
l= 0,
, n), where (7)
l=l,
,n
(8)
The main reason for the different results m Proposltlons 1 and 2 1s that m G( 1, n) land units are substitutes for the developer, while m G(n, n) land units are complements The developer, hence, has a number of alternative choices for the development m G( 1, n) for n 2 2, which 1s not the case m G(n, n) The two cases considered above are very special cases, where the order of negotiations does not change the developer’s share If 2 5 k sn1, however, the order of negotiations can be a very important strategic variable for the developer Consider G(2,3) In this case, the developer must buy landowner 2’s land unit, but he needs only one of the peripheral land units In other words, player 2 has a stronger bargaining position than players 1 or 3 Intmtlvely
Y Asaml and A Terakl, Optrmal negotlatlon orders and land prices
543
the developer appears to be better off by begmnmg the negotlatlon with player 2, because it would appear that landowners m the earher negotlatlons have weaker bargaining posltlons Surpnsmgly, this mtultlon 1s false, as 1s shown m Proposltlon 3 The order of negotlatlons does not affect player O’s share, although player 2’s share 1s actually lower d negotiations are held with him first Proposztzon 3 In G(2,3), the order of negotzatzons does not affect player O’s share If negotzatzons are held wzth player 2 first, then hzs share IS 318, otherwise l/2 The expected payoff IS given by (318, l/12,1 l/24,1/12) In Proposltlon 3, it 1s observed depend on the order of negotlatlons property holds for more complicated to consider the case of G(2,4) As 1s order becomes an important strategic
that the developer’s payoff does not Hence one may ask whether this situations In particular, it 1s of interest shown m the followmg proposltlon, the variable for the developer m this case
Proposztzon 4 In G(2,4), player O’s optzmal negotzatzon orders are all those whtch ezther (I) start from a peripheral landowner (I e , 1 or 4), or (II) start from a central landowner (z e 2 or 3) and proceed to an adjacent landowner In all cases, player O’s share IS l/2, and the expected payoff zs gzven by (l/2,7/80, 13/80,13/80,7/80) One example of a non-optimal negotiation order 1s [3,1,4,2] This 1s mtultlvely understandable since player 0 will waste money by buying land unit 1, which 1s never used m the development With this m mmd it might appear surprlsmg that the order [4,1,3,2] 1s optimal, since m this order, player 0 will also waste money by buying land unit 1 One different aspect m this order, however, 1s that by negotiating with player 1, player 0 succeeds m reducing player 4’s negotiation power In particular, if player 4 does not agree, then the remaining order [l, 3,2] 1s still an optimal negotiation order for the subgame that player 4 1s excluded from the orlgmal game Stated differently, all the remaining land units are still effectively available for the remaining negotiation Hence player 0 has a higher status-quo position for the first negotiation In the former order, however, d player 3 rejects the land procurement, then the remaining order [1,4,2] 1s not optimal for one land unit can never be used for the development even though there are three remaining land units Player 0, hence, has a worse status-quo position m the first negotiation These observations are consistent with the mtultlve prmclple that the negotiation order should be so selected that the remaining negotiation order at each step continues to be advantageous If we fix the order of negotiation as [l, 2, , n], then it 1s possible to calculate the shares for the cases of G(2,n)
Y Asamr and A Terakl, Optimal negotlatlon
544
Proposmon 5 In G(2, n), If player share p”, IS given by
0 adopts
orders and land prrces
the order
[l, 2,
,n],
then the
p;=I-3v,_1-v,_2,
(9)
(11) p:=O,
for
1>3,
(12)
where Uk
=([(l + &)/41k-[(1-$)/41k)i(2&)
4. An alternatwe
(13)
model
So far it has been assumed that the developer must pay all prices agreed upon, even If he falls to assemble a feasible set of land umts for development Smce there IS only one developer m the model, It may be argued that this IS too restrlctlve an assumption If we Interpret a prior commitment to buy land as d way to reserve this land, then a developer does not necessarily have to reserve land if there IS no other competitor To modify this sltuatlon, we adopt the alternatlve assumption that the developer has the right not to pursue the project and cancel all prices (1 e , all contracts to buy) However, it will be assumed that if the developer proceeds with development, then he must pay prices not only to the landowners of relevant land units, but also to all other landowners with whom he has agreed to buy land units ’ Based on this ‘condltlonal contract’ assumption, we must modify all the results For a notational convenience, this alternative game ~111 be designated as G’(k, n) Proposltlon share are (E, I =O,
6
In G’(1, n), player
and , n), where
1 -l/2”
l/2”,
O’s share
respecmely
and the first The
expected
negotiated payoff
player’s
IS given
by
‘We can consider stdl another scheme m winch the developer has a right to cancel buying the land for any subset of land units However m such a case, we may face some mdetermmacy problems Consider for example G( 1.2) If the developer selects the order [1,2] and lf the price of land umt 1 IS settled as X, then the price negotlatlon with landowner 2 wdl result m d price of u/2 Now the questlon IS how to determme the value of x The status-quo IS (l/2,0,*), and money left for negotlatlon 1s 1 But then IS Y equal to l/27 It seems that the Nash bargammg solution concept Itself IS Irrelevant here, for the result of the second negotlatlon cannot affect the solution Hence, to deal with such a case, it appears that we must take a non-cooperative extensive-form game approdch
Y Asamr and A Terakz, Optimal negotlatlon orders and land prrces
E, = 1 - l/2”, E,= l/(n2”),
545
(14) I= 1,
,n
(15)
It 1s worth noting that the result m Proposltlon 6 1s exactly the same as m Proposltlon 1 In other words, if k= 1, then shares are not affected by whether the developer has a right to cancel buymg commitments The reason 1s that the developer’s status-quo position never becomes negative even m G( 1, n) If kz 2, however, this modlficatlon of the assumptions crltlcally affects the result Proposztzon 7 share If [1,2,
In G’(n,n), the order of negotzatzons does not affect player O’s , n] zs adopted, the share, p, IS gzven by
p,=l/(n+l),
z=O,l,
,n
The expected payoff IS gzven by (E, z=O, E,=l/(n+l),
z=O,l,
,n
(16) , n), where (17)
It IS somewhat surprlsmg that the order does not affect any player’s share This 1s due to the fact that the developer does not have to make a commitment to buy By this assumption, the status-quo position for each player 1s zero m any negotiation, and hence the profit 1s shared equally In G(2,3), it was observed that the order of negotiations does not affect player O’s share The followmg proposltlon shows that this 1s no longer valid m G’(2,3) Proposztzon 8 In G/(2,3), the optzmal order of negotzatzons for player 0 IS to start the negotzatzon wzth player 1 or 3 Then players 0 and 2’s shares are the same and equal 419 The first negotzated landowner’s share IS l/9 The expected payoff IS gzven by (419, l/18,4/9,1/18) It seems quite uruntultlve that the negotiation order starting with player 2 IS not optimal In this game player 0 and player 2 get the same share m all cases, for the status-quo 1s zero m any case for both players (which 1s the mam difference from the original model) Then the optimal order must involve some orders which weaken player l’s (or 3’s) negotiation power This 1s the reason why the developer should start negotiation with player 1 (or 3) One may wonder why both [l, 2,3] and [l, 3,2] are both optimal m G’(2,3) Suppose first that the first negotiation falls Then It 1s evident that both orders are optimal m the remaining negotiation Now suppose that the developer decides to buy land unit 1 In this case, whether the development
546
Y Asamr and A Terakl, Optrmal negotlatlon
orders and land prices
project can be carried out or not depends totally on the negotiation with player 2 In other words, buying land unit 3 does not help the developer at all Since the developer can freely decline to buy land m any negotiation (or he can mslst on buying the land with price zero), [ 1,2,3] and [l, 3,2] are both equivalent to [l, 23 provided that the developer buys land unit 1 with a price under [l, 2,3] Note that m G/(2,3) the developer 1s better off by first negotiating with player 1 or 3 Hence one may ask whether this property holds for more complicated cases To allow a meaningful comparison, let us consider the game G’(2,4) Proposltzon 9 In G’(2,4), player O’s optimal negotlatlon orders are all those starting from a peripheral landowner (1 e, 1 or 4) followed by an adjacent landowner (1 e, 2 or 3) In all cases player O’s share 1s 26145, and the expected payoff 1s given by (26145, l/l 5,13/90,13/90,1/l
5)
The optimal orders are [1,2,3,4], [1,2,4,3], [4,3,2, l] and [4,3,1,2] These orders satisfy the condltlons that (1) even if the first negotiation falls, the rest of the order 1s optimal m the remaining subgame, and (11) the developer does not have to pay a price for any unnecessary land unit Hence the basic principle suggested by the model m section 3 above 1s still valid One common property observed m both models 1s that the optimal order of negotiation should be so determined that if the first negotiation falls, then the remaining order continues to be optimal m the remaining subgame If we employ this prmclple recursively, then it follows that for any negotiation which falls, the remammg order continues to be optimal This prmclple, which can be called ‘maximum principle of negotiations’ (or a ‘subgame optimal prmclple’), has an mtultlve appeal For if the developer follows this prmclple, then he can maintain the strongest status-quo position m any negotiation To state this principle more precisely, let O(m) be the set of all possible permutations of [l, ,m], which denotes all the possible negotiation orders Define the set of optimal orders, o*(m), such that for any where n(0) 1s the developer’s share c7E O*(m) c O(m), n(a) =mdxoE 8(mj n(O), associated with the order 0 Take CJEO(m) and z E O(m + 1) If CJand 7 satisfy ,x,,,+J 1s the same as [a,, that [z,, ,oml or Co,+ 1, a,+ 11, then ~7 IS said to be embedded in T Define the set of orders, @‘(mil), m O(m+ 1) m which some optimal order m O*(m) 1s embedded O+(m+
1) = (0~ O(m + 1) there exists (TE O*(m) embedded
With this notation,
the prmclple
such that B 1s
m 0) above
1s convemently
expressed
as
Y Asamr and A Terala, Optimal nqotratron orders and land pees
541
Maximum Prrnclple of Negotlatron max ZZ(O)= max Z7(cr) aEf3+(a-l) BE6W Though It IS not proved here,’ the order [l, 2, ,n] seems to be optlmal for G’(2,n), for this order sat&es the principle above If we fix the order of , n] (which 1s an optimal order for n= 2, 3 and 4), negotiations as [ 1,2, then It 1s possible to calculate the shares for the case of G’(2,n) Proposltlon 10 In G’(2,n), If player 0 adopts the order [1,2, share p” IS gzven by
,n], then the
P”o= Yll?
(18)
P;=(1-2Y”-,+V”-J/(3+Y,-J,
(19)
P);‘(l
(20)
+Y,-,)(~-Y”-,)/(3+Y”-,)~
pf=O
for
(21)
123,
where y, IS recursively defined by y,=(l
+y,-I)(1
+y,-,)/(3+y,-2)
(22)
and (23)
Yo=Y,=O Moreover these terms satisfy lim p;= Il’a)
1,
(24)
lim py = lim p; =O, n-CC n-m
(25)
llm (P;/P;)=($“-CC
I)/4
(26)
The result that the developer’s share approaches 1 as n Increases IS quite understandable For n large, even if the developer falls m the first negotlatlon, he still has a lot of other possible assemblies of land umts, which strengthens his bargaining position accordingly It 1s of interest to note that ‘At a first glance, It may not appear a dlffkult problem argument The proof, however, will mvolve a combmatorlal
to prove by a mathematical mductlon analysis, which IS hard to conduct
548
Y Asam and A Terakl, Optrmal negotlatlon orders and land pruzes
h-n, + (P;/P;)=(&-
I)/4 Th IS means that when n-+ co, player 2’s land umt 1s worth (1 + a) times that of player 1 (even though the absolute value of prices approaches zero) Since player 2’s land unit 1s necessary m order to develop player l’s land unit, it 1s easily seen that player 2 should obtain a larger share than player 1 The proposition says further that even if n goes to mfimty the ratio between these prices never goes to zero 9 m
5. Extensions
to various land unit sizes
So far, it has been assumed that all landowners possess land units of the same size In this section, this assumption 1s relaxed to include cases m which landowners own various sizes of land units Toward this extension, several modliicatlons m notation will be introduced m the followmg Let a be the size of land mmlmally necessary to yield a fixed profit 1 to the developer (called player 0) It 1s assumed as before that the developer can gam only one unit of profit by developing a size of land larger than or equal to a There exist n landowners Each landowner, 1, possesses a land unit of size S, All land units he on a line In particular, landowner I owns all land m the interval, CC; = 1 S, - S,, It, = 1 S,), m the total land of [O,cJ= 1 S,) Since this extended version of land development game can be characterized by the necessary land size, a, and all landowners’ land unit sizes, S=(S, J = 1, I 4, the model 1s designated ds g(u,S) hereafter One might argue that if a m the landowner possesses a larger unit, then he has an advantage negotiation of land price, when the size of land unit varies among landowners To examme this mtultlon, we begin with an analysis using assumptions similar to that m section 3 Suppose again that the landowner cannot cancel the negotiated prices The other assumptions on the negotiation process are the same as m section 3 To simplify the argument, let us introduce the following notation Let L(~,J) be the set of landowners ordered from I to J, where 1_I J L(t,J)={t,
,J1-
(27)
Then the family, L, of mmlmal developer can succeed to develop follows
L=
L(L,J)
91t can be readdy
i
k=,
S,Za,
i k=t
sets of landowners, land to obtain profit
Sk-S,
i
with whom the land 1, can be expressed as
Sk-S,
k=r
shown that tlus hrmt 1s the same for the basx
3
model above
Y Asamzand A Terakl, Optimal negotlatton orders and land prrces With this notation, it 1s shown that if n=3, and rf a landowner included m any mmlmal set m the family, L, then his land price irrespective of the order of negotiation Proposztzon 11 In g(a,S) wzth S=(S, zn L, then J’S share, p,, zs zero
I= 1,2,3),
If
J IS
549
IS not 1s zero
not included zn any L
Proposltlon 11 implies that d a landowner’s land unit 1s not necessary for the development, then he is practically ignored m the negotiation The Shapley value [Shapley (1951, 1953)‘J 1s often used to find the relative negotiation powers among players It 1s seen that the share m our negotlatlon game, and the Shapley value exhibit a slmllar feature In particular, a large landowner has an advantage m the negotiation only if the required land size, a, 1s relatively small Proposztzon 12
In g(a, s) wzth S= (S, l= 1,2,3)
If asmax, S,, then the landowner who owns the largest land unzt obtazns the largest Shapley value among all landowners (II) If a>max,S,, then landowner 2’s Shapley value zs the largest among all landowners (I)
The proposltlon above means that if the required land size, a, 1s smaller than the largest land unit, then the share of the largest landowner 1s the largest of all landowners, and if the required land size 1s larger than the size of the largest land unit, then the developer has to negotiate with several landowners and thus the middle landowner receives the largest share As m the alternative model m section 4, we can also consider an alternative extended model with a variety of land sizes, m which the developer preserves a right to cancel all the negotiated results at the end We denote this extended model by g’(a,S) Then we can analyze optimal orders of negotiation as follows Proposztzon 23
In g’(a,S) wzth S=(S,
z= 1,2,3)
Zf max(S,,S,)
In the alternative
extended
model, g’(a,S),
if S=(S,,S,,S3)
satisfies
one of
the condltlons m Proposltlon 13, then there exist (m) above reduces to G’(2,3), for the developer combmatlons of land units, il, 2) or (2,3} On developer can choo,se between one larger un:t and and (ii) above], then it is ddvantageous for him to landowner first
optimal orders The case can choose one of two the other hand, d the two rmall unit3 [l e, (1) negotiate with one large
6. Concludmg remarks
This paper has analyzed a simple class of sequential negotiation games, m which one developer negotiates land prices with landowners m order to develop a project worth one unit of profit It was shown m Proposltlon 1, that if the developer has n alternative plots avdilable, then his share approaches 1 as II increases The ‘converse’ case was tredted m Proposition 2 If the developer needs all the n land units, then his share approaches 0 as n increases It was shown that the order of negotlatlons may or may not be a strategic variable even If the developer has some alternatives of development patterns each of which consists of several land units In particular, m G(2,3) the order does not affect player O’s share (m Proposltlon 3), while m G(2,4) it does (in Proposltlon 4) The basic prmclple seems to be that player 0 should select the order so that he can retam an optimal order after the first negotiation An alternative model was analyzed rn section 4 m which the developer retains a right to cancel all prices It was shown m Proposltlon 6 that if the developer needs only one plot, then the result remains the same as m the former model If the developer needs all the plots, however, then the players equally divide the potential profit (m Proposltlon 7) In G’(2.3) and G’(2,4), the order of negotlatlons become a crItica strategic variable for the developer In both cases, it was observed that an order which satisfies the ‘maxlmum prmclple of negotlatlon’ if optimal This principle, namely that ‘the order of negotlatlons should be selected so that even if any negotiation falls, the remammg order 1s still optimal m the remammg subgame’, appears to be a fundamental rule for negotiation processes In section 5, an extension of the basic model 1s analyzed m which the sizes of land units are allowed to vary It was shown in Proposition 11 that the price of ‘redundant’ land units which does not influence the posslblhty of development is dlwdys zero Moreover, it is optimal for the developer to start the negotiation with the landowner with the largest land umt provided that the developer hds the choice between one large unit and two small units (Proposition 13) A very important class of models are left for future research, namely models which allow for collusion of players and/or renegotiation In our models, for example, if the developer and landowner collude secretly, then it
Y Asamr and A Terakr
Optlm&l negotlatlon
orders and land prwes
551
may benefit both players If this 1s the case, then some players may suspect the existence of a secret collusion Such a model with Imperfect information 1s a new lme of research to be conducted m the future Even wlthout the posslblhty of any secret collusions, we may derive a different result by taking the posslblhty of open collurlons mto account On the other hand, If we permit renegotlntlon, then we may derive a more even dlstrlbutlon of share Ii)
Appendix: Proofs of propositions A I
Proof
of Proposltlon
1
The argument 1s by mathematical mductlon For n= 1. the result follows from section 2 Moreover, if the assertlon holds for all ni k, then, for G( 1, k + l), the status-quo point of the first negotlatlon 1s (1 - 1/2k. 0, *. *), where the negotlatlon order 1s [1,2, ,n] Hence it follows that pi = 1 - 1/2kf1 and p1 = l/2 k+’ Fmally noticing that the order does not matter for the developer, the result follows’ QED A 2
Proof of Proposltlon
2
Again, employing mathematical mductlon, observe that for n= 1 and 2, the result follows from yectlon 2 Moreover, If the assertion holds for all n < k. then m G(k + 1. k+ 1) the status-quo point of the first negotiation 1s , *), where the negotiation order 1s [1,2, ,n] Since other players (0,0, *, (than players 0 and 1) will get the total share of 1- lj2k, only 1/2k 1s left for both to share Hence it follows that p0 = 1/2k+1 and p1 = 1/2k+1 Finally noticing that the order does not matter for player 0, the result follows QED A 7 Proof of Proposltlon
3
If the order is [2,1,3], then the share is (3/8, l/4,3/8,0) If the order IS [1,2,3], then the share 1s (3/8, l/8,1/2,0) If the order 1s [1,3,2], then the share 1s (3/S, 1/8,1,‘2,0) All other cases can be verified similarly QED A 4
Proof of Propontlon
4
If the order is [1,2,3,4], then the share 1s (l/2, l/S, 3/8,0,0) If the order 1s Cl, 3,2,4], then the share 1s (l/2, l/8,1/4,1/8,0) If the order 1s [2,1,3,4-J, then the share 1s (l/2, l/4,1/4,0,0) If the order 1s [2,4,1,3], then the share 1s (7/l 6, l/4,3/16,0, l/8) All other cases can be verified similarly QED
‘?ke Asaml
(1988) for a study along thts hne
Y Asaml and A Terakz, Optimal negottatlon orders and land prrces
552
A 5
Proof of PropoAltlon
5
For n = 2,3, the py’s are given above For larger relationship among p& py and p; for each n P”o’ l
Solvmg
A 6
values
of n, we have the
=(l +p”o-‘+2p1;)/4,
this system with the mltlal
Proof of Proposctlon
condltlons,
we have the result
QED
6
For n= 1, the status-quo 1s (0,O) and thus pO=pl = l/2 assertion holds for all ns k, then m G’( 1, k + 1) at the first status-quo IS (1 - 1/2k, 0, *, , *) given the negotiation order it follows that pO= 1 - 1/2k+1 and p1 = 1/2k+’ Since the matter for the developer, the result thus follows by induction
A 7 Proof of Proposltlon
Moreover, if the negotiation, the [l, 2, ,nl, and order does not
QED
7
For n= 1, p, = l/2 for I =O, 1 If the assertion holds for n 5 k- 1, then m *) given negotiaG’( 1, k) at the first negotiation, the status-quo 1s (0,0,*, , k] Moreover, if r IS the price of lani unit 1, then all tion order [l, 2, players other than players 0 and 1 will get a total share of (k- 1) (1 -r)/ (k+ 1) Hence only r +( 1 - k)/k 1s left for players 0 and 1 to share, and it follows that r =(r +( 1 - r)/k)/2 or r = l/(k + l), which implies that p, = l/(k + 1) , k Again noticing that the order does not matter for player 0, the for 1=0, result follows by mductlon QED
A 8
Proof of Proposltlon
8
If the order 1s [2,1,3], then the share 1s (3/7, l/7,3/7,0) If the order 1s [l, 2,3], then the share is (4/9, l/9,4/9,0) If the order 1s [l, 3,2], then the share IS (4/9, l/9,4/9,0) Other cases can be calculated similarly, and the result follows QED
A 9
Proof of Proposltlon
If the order
9
1s [ 1,2,3,4],
then
the share
1s (26/45,2/15,13/4&O,
0) If the
Y Asaml and A Terakl, Optrmal negotratlon orders and land prwes
order 1s order IS order 1s [2,4,1,3], slmllarly
A 10
553
[1,3,2,4], then the share 1s (90/161,3/23,30/161,20/161,0) If the [1,4,2,3], then the share 1s (13/23,25/207,13/69,0,26/207) If the [2,1,3,4-J, then the share IS (4/7,4/21,5/21,0,0) If the order 1s then the share is (12/23,4/23,13/69,0,8/69) All other cases can be verified QED
Proof of Proposmon
10
For n=2, 3 and 4, the proposltlon follows from Propositions 7, 8 and 9 Moreover, if the assertion holds for all n 2 k- 1 (2 4), then m G’(2, k) if the first negotiation falls, the remaining case reduces to G’(2, k- 1) Thus the status-quo of player 0 1s y,_ 1 Suppose the first negotiation results m a price of x If the second negotiation falls, then the remaining case reduces to G’(2, k-2) with a total payoff of (1 -x) Hence it follows that the status-quo of player 0 m the second negotlatlon 1s given by (1 -x)y,-,, so that the total share left for players 0 and 2 1s 1 -x Hence players 0 and 2’s shares are (1 -x) (1 + y, ~ J2 and (1 -x) (1 - y, _ J2, respectively In the first negotlatlon, the status-quo of player 0 1s y,_ 1 as shown above The total share which can be shared by players 0 and 1 1s 1 - (1 -x) (1 -y,_ J/2 It follows that x=(1-2yk_,+yk_,)/(3+y,_,) Hence
If O
have It
we
[r._1(4-z,_,)]
+J5)+(,/5-3)z,_J4]/ Ir,-(1+,/‘$/4~=hm,,,
(r,_,-
+Js) hm,+, r,-J
(1+$)/4I/C(l
lmplymg (2 -z,
r--(1 +J5)/4=[1-4r,_,/(l follows that hm,,,
that
554
A I1
Y Asamr and A
Terukr,
Proof of Proposltlon
Optrmal
negotratlon
order,
and land prrc~\
11
(a) If S, 5 S, 5 S, holds, then If UPS,, then everyone IS Included m some L m L, If S,
(b) If S, 5 S, 5 S, holds, then (1) if ulS,, then everyone IS included m some L m L, (II) if S,
A 12
cases
can
be
Proof of Proposltlon
12
(a) If S, 5 S, 5 S, holds,
(I) if uzS,, then (11) if S,
verified
similarly
Thus
the
proposltlon
1s
QED
then
the Shapley value 1s given by (3/4, l/12, l/12, l/12), then the Shapley value IS given by (2/3,0,1/6, l/6), and UPS, +S,, then the Shapley value 1s given by (7/12, and
S, +S,
then
the Shapley
the Shapley value
value
IS given
I$ given
by (l/2,
by (5j12, l/12, l/12,
then the Shapley value IS given by S,+S,
Y Asaml and A Terakl, Optrmal negotratron orders and land prvzes
(b)
555
If S, 5 S, 5 S2 holds, then
If d (111)if (14 If
II) (4 (VI)
asS,, then the Shapley value 1s given by (3/4, l/12,1/12, l/12), S, SusS,, then the Shapley value IS given by (2/3,0,1/6, l/6), S, CUSS,, then the Shapley value 1s given by (l/2,0, l/2,0), S,
l/l 2), If S,
IS given
by
by (l/4, l/4,
l/4,1/4) (c) All other cases can be verified result QED A 13
Proof ofPropos&on
similarly,
and hence
we have the desn-ed
13
The proposltlon can be verified by exammmg the proof of ProposItIon 11 QED
all cases exhaustively
as m
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