Reglonal Science and Urban Economics 20 (1990) 537-556 North-Holland

ON SEQUENTIAL NEGOTIATION PROCEDURES

Optimal Negotiation Orders and Land Prices*

Yasushl ASAMI

Unrterslty of Tokyo, Tokyo, Japan

Aklhlro TERAKI

Burldmg Research Institute, Mzmstry of Constructmn, Tsukuha, Japan

Received 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 develop- ment 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 three- player/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 and Txole (1983), Roth and Schoumaker (1983), Rubmstem (1985), Rubmstem and Wohnsky (1985) and Hailer (1986)

‘See also RIddell ( 198 1)

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 More- over, 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 orders and land prrces

each zth landowner owns the single land unit, [l- 1, I), m the interval [0, n) Thus the land market consists of n contiguous land units The developer 1s denoted by player 0, and the lth landowner 1s denoted as player I Any feasible payoff (share), p E R”+ ‘, must satisfy the 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 negotla- tlon, 1 e, 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 status- quos 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 m the current analysis In such a case, ‘t’ 1s used to denote the correspondmg element

542 Y Asamr and A Terakl, Opttmal negotlatron orders and land prxes

land umt One expects that as the number of landowners Increases the developer’s payoff approaches the total profit 1 This expectation IS con- firmed 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”, (3)

E,=l/(n2”), l= 1, ,n (4)

Next consider a sltuatlon m which the developer needs all the land umts In this case, it 1s shown below that the later a landowner appears m the order of negotlatlons, the larger 1s the share he obtams Moreover, the developer’s share approaches 0 as the number of landowners increases

Proposltlon 2 In G(n,n), the order of negotlatlons does not affect player O’s

share If [l, 2, ,n] IS adopted, the share, p, 1s given by

PO = l/2”, (5)

pl=1/2”+‘-‘, r=l, ,n (6)

The expected payoff 1s given by (E, l= 0, , n), where

E, = l/2”, (7)

E,=(l-1/2”)/n, 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 sn- 1, 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 that the developer’s payoff does not depend on the order of negotlatlons Hence one may ask whether this property holds for more complicated situations In particular, it 1s of interest to consider the case of G(2,4) As 1s shown m the followmg proposltlon, the order becomes an important strategic 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 prm- clple 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)

544 Y Asamr and A Terakl, Optimal negotlatlon orders and land prrces

Proposmon 5 In G(2, n), If player 0 adopts the order [l, 2, ,n], then the

share p”, IS given by

p;=I-3v,_1-v,_2, (9)

where

p:=O, for 1>3,

Uk =([(l + &)/41k-[(1 -$)/41k)i(2&)

(11)

(12)

(13)

4. An alternatwe 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 6 In G’(1, n), player O’s share and the first negotiated player’s

share are 1 -l/2” and l/2”, respecmely The expected payoff IS given by (E, I =O, , n), where

‘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 545

E, = 1 - l/2”,

E,= l/(n2”), I= 1, ,n

(14)

(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 In G’(n,n), the order of negotzatzons does not affect player O’s share If [1,2, , n] zs adopted, the share, p, IS gzven by

p,=l/(n+l), z=O,l, ,n (16)

The expected payoff IS gzven by (E, z =O, , n), where

E,=l/(n+l), z=O,l, ,n (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 c7E O*(m) c O(m), n(a) =mdxoE 8(mj n(O), where n(0) 1s the developer’s share associated with the order 0 Take CJE O(m) and z E O(m + 1) If CJ and 7 satisfy

that [z,, ,x,,,+J 1s the same as [a,, ,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) such that B 1s

embedded m 0)

With this notation, the prmclple above 1s convemently expressed as

541 Y Asamr and A Terala, Optimal nqotratron orders and land pees

Maximum Prrnclple of Negotlatron

max ZZ(O) = max Z7(cr) BE 6W aEf3+(a-l)

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 negotiations as [ 1,2, , n] (which 1s an optimal order for n= 2, 3 and 4), 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

P”o = Yll?

P;=(1-2Y”-,+V”-J/(3+Y,-J,

P);‘(l +Y,-,)(~-Y”-,)/(3+Y”-,)~

pf=O for 123,

where y, IS recursively defined by

y,=(l +y,-I)(1 +y,-,)/(3+y,-2)

and

Yo=Y,=O

Moreover these terms satisfy

lim p;= 1, Il’a)

lim py = lim p; =O, n-CC n-m

llm (P;/P;)=($- I)/4 “-CC

,n], then the

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(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 negotla- tlon, 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 to prove by a mathematical mductlon argument The proof, however, will mvolve a combmatorlal analysis, which IS hard to conduct

548 Y Asam and A Terakl, Optrmal negotlatlon orders and land pruzes

h-n, + m (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

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 landowner possesses a larger unit, then he has an advantage m the negotiation of land price, when the size of land unit varies among land- owners To examme this mtultlon, we begin with an analysis using assump- tions 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 sets of landowners, with whom the land developer can succeed to develop land to obtain profit 1, can be expressed as follows

L= L(L,J) i S,Za, i Sk-S,<a,and i Sk-S,<a k=, k=t k=r 3

91t can be readdy shown that tlus hrmt 1s the same for the basx model above

Y Asamz and A Terakl, Optimal negotlatton orders and land prrces 549

With this notation, it 1s shown that if n=3, and rf a landowner IS not included m any mmlmal set m the family, L, then his land price 1s zero irrespective of the order of negotiation

Proposztzon 11 In g(a,S) wzth S=(S, I= 1,2,3), If J IS not included zn any L zn L, then J’S share, p,, zs zero

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 negotla- tlon 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)

(I) 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

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)

(1) Zf max(S,,S,)<a5S, and a5S,+S,, then zt IS optzmalfor the developer to start negotzatzon wzth landowner 3

(II) Zf max(S,,S,) <ass, and a SS, +S,, then zt IS optzmal for the developer to start negotzatzon wzth landowner 1

(zzz) If max, S,<a, ass, +S,, and ass, +S,, then zt IS optzmal for the developer to start negotzatzon wzth a perzpheral landowner (I e , 1 or 3)

(zv) If S does not satzsfy any condztzons above, then the share of the developer does not depend on the order of negotzatzon

In the alternative extended model, g’(a,S), if S=(S,,S,,S3) satisfies one of

the condltlons m Proposltlon 13, then there exist optimal orders The case (m) above reduces to G’(2,3), for the developer can choose one of two combmatlons of land units, il, 2) or (2,3} On the other hand, d the developer can choo,se between one larger un:t and two rmall unit3 [l e, (1) and (ii) above], then it is ddvantageous for him to negotiate with one large landowner first

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

(0, 0, *, , *), where the negotiation order 1s [1,2, ,n] Since other players (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

552 Y Asaml and A Terakz, Optimal negottatlon orders and land prrces

A 5 Proof of PropoAltlon 5

For n = 2,3, the py’s are given above For larger values of n, we have the relationship among p& py and p; for each n

P”o’ l =(l +p”o-‘+2p1;)/4,

Solvmg this system with the mltlal condltlons, we have the result QED

A 6 Proof of Proposctlon 6

For n= 1, the status-quo 1s (0,O) and thus pO=pl = l/2 Moreover, if the assertion holds for all ns k, then m G’( 1, k + 1) at the first negotiation, the status-quo IS (1 - 1/2k, 0, *, , *) given the negotiation order [l, 2, ,nl, and it follows that pO= 1 - 1/2k+1 and p1 = 1/2k+’ Since the order does not matter for the developer, the result thus follows by induction QED

A 7 Proof of Proposltlon 7

For n= 1, p, = l/2 for I =O, 1 If the assertion holds for n 5 k- 1, then m G’( 1, k) at the first negotiation, the status-quo 1s (0, 0, *, *) given negotia- tion order [l, 2, , k] Moreover, if r IS the price of lani unit 1, then all

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) for 1=0, , k Again noticing that the order does not matter for player 0, the

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 9

If the order 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 553

order 1s [1,3,2,4], then the share 1s (90/161,3/23,30/161,20/161,0) If the order IS [1,4,2,3], then the share 1s (13/23,25/207,13/69,0,26/207) If the order 1s [2,1,3,4-J, then the share IS (4/7,4/21,5/21,0,0) If the order 1s [2,4,1,3], then the share is (12/23,4/23,13/69,0,8/69) All other cases can be slmllarly verified QED

A 10 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 negotla- tlon, 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<y,_,, y,_,<l, then O<y,<l by eq (22) Define z,=l-y, Then

z,=(22,_1 +z,_z -z,-1z,-2)/(4-z,-2) It follows that z,-z,_1= -(2z,_,-z,_,)/(4-z,~,) If Oj~,_~<z,_~~l, then O~Z,<Z,~~ Since 0~ z,tz,< 1, it follows that {z,> 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~_~<l, then 1~r,=(2+(1-z,_,)/r,_,)/(4-z,_,)~(3-z,_,)/(4-z,_,)~1/2 Since r,=5/ 6, it follows that for any nL3, 1/2<r,<l Smce rnrn-1(4-z,_2)=2r,_1 +l -

z,-~, we have r--(1 +J5)/4=[1-4r,_,/(l +J5)+(,/5-3)z,_J4]/

[r._1(4-z,_,)] It follows that hm,,, Ir,-(1+,/‘$/4~=hm,,, (r,_,-

(1+$)/4I/C(l +Js) hm,+, r,-J<W~1 +J5)) llm,+, Ir,-, 4 +JJW(, lmplymg that hm, _ o. r, = (1 + $)/4 hm, _ m (p”‘/p”‘) = hm,, ~ (2r, _ 1 - l)/

(2 -z, _ 1) = (fi - 1)/4 Hence the proposition is established QED

554 Y Asamr and A Terukr, Optrmal negotratlon order, and land prrc~\

A I1 Proof of Proposltlon 11

(a) If S, 5 S, 5 S, holds, then

(I) If UPS,, then everyone IS Included m some L m L, (II) If S, <uzSz, then player 1 1s not included m any L 111 L and pI =O, (in) if S, <asS, and as.S, +S,, then everyone IS included 111 some L m L, (IV) if .!?,<a SS, and S, +S? <a, then players 1 and 2 are not included m

any L m L and p, =pz=O, (v) if S, <u 5 S, + S,, then everyone 1s included m some L 1x1 L. (VI) if S, <u and S, +S, <(I 5 S, + S,, then player 1 1s not included m any L

m L and p, =O, and (vu) If S, < a and S, + S 3 <a, then everyone 1s included m some L III L

(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 5 S,, then player 1 is not included m any L In I, and p, = 0, (in) if S, <a $Ss,, then players 1 and 3 are not included m any L m I, and

p1 =p3=0, (iv) If S, <u 5 S, + S,, then everyone 1s included m some L in L, (v) if S, < u and S, + S2 <us S, + S,, then player 1 IS not included m any L

m L and p1 =O, and (vi) if S, <u and S, + S, <(I, then everyone IS included m some L m L

(c) All other cases can be verified similarly Thus the proposltlon 1s established QED

A 12 Proof of Proposltlon 12

(a) If S, 5 S, 5 S, holds, then

(I) if uzS,, then the Shapley value 1s given by (3/4, l/12, l/12, l/12), (11) if S, <uzS,, then the Shapley value IS given by (2/3,0,1/6, l/6), (m) if S,<u ZS, and UPS, +S,, then the Shapley value 1s given by (7/12,

l/12, l/12,1/4), (IV) if S, <a SS, and S, +S, <a, then the Shapley value I$ given by (l/2,

O,O, l/2), (v) if S,<usS,+S,, then the Shapley value IS given by (5j12, l/12, l/12,

5/l2), (vi) if S,<u and S,+S,<ugSS,+S,, then the Shapley value IS given by

(l/3,0,1/3,1/3), and (vu) if S, <a and S, + S, <a, then the Shapley value IS given by (l/4, l/4, l/4,

l/4)

(b)

II) (111) (14

(4

(VI)

Y Asaml and A Terakl, Optrmal negotratron orders and land prvzes 555

If S, 5 S, 5 S2 holds, then

If asS,, then the Shapley value 1s given by (3/4, l/12,1/12, l/12), d S, SusS,, then the Shapley value IS given by (2/3,0,1/6, l/6), if S, CUSS,, then the Shapley value 1s given by (l/2,0, l/2,0), If S,<a~Sl+S2, then the Shapley value IS given by (5/12, l/12,5/12,

l/l 2), If S, <a and S1 +S, CUSS, + S,, then the Shapley value IS given by

(l/3,0, l/3, l/3), and if S, <a and S2 +S, <a, then the Shapley value IS given by (l/4, l/4,

l/4,1/4)

(c) All other cases can be verified similarly, and hence we have the desn-ed

result QED

A 13 Proof ofPropos&on 13

The proposltlon can be verified by exammmg all cases exhaustively as m the proof of ProposItIon 11 QED

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