Optimization and game theoretic modeling of the real estate development Текст научной статьи по специальности «Математика»

Gennady A. Ougolnitsky

Southern Federal University,

Department of Applied Mathematics and Computer Sciences, Faculty of Mathematics, Mechanics and Computer Sciences, Milchakov St., 8A, Rostov-on-Don, 344090, Russia E-mail: ougoln@mail.ru

Abstract A system of the real estate development optimization and game theoretic models is described. A basic role in the proposed system is played by aggregate models of a real estate development company. Those ones are static optimization models aimed at the definition of optimal prices with constraints on the solvent demand. A natural generalization of the basic model is possible in two directions: “horizontally” and “vertically”. First, an interaction of real estate development companies as equal economic agents may be considered. In turn, two model approaches are possible in this case. If we consider competitive relations of development companies without formation of coalitions then non-cooperative games of n players in normal form arise.

If a cooperation is admissible (common resources, mergers and acquisitions of development companies) then we get cooperative games. Second, development companies have economic relations with organizations of other types.

These relations are hierarchical as a rule, and a development company can be both a Leader (in relations with its suppliers) and a Follower (in relations with its investors, credit institutions, administrative agencies). Respectively, hierarchical game theoretic models arise.

Keywords: game theory, optimization models, real estate development.

1. Introduction

A real estate development (RED) project provides implementation of the whole cycle of investments in capital construction works. The subject of the activity is a real estate development company, or developer.

Real estate development objects are subdivided into types and classes. The main types are urban and suburban residential real estate, commercial real estate, trade and entertainment centers, stores, industrial buildings. The principal classes are premium class (A), business class (B), economy class (C), social class (of the residential real estate as a rule). Intermediate classes as B+ ,B-and so on are also possible. There are professional classifiers which define the class of an object. The class of an object determines the productive costs and possible prices. Each RED-object may be characterized by an index which combines its type and class.

It is evident that there are many developers on each territory. Their “horizontal” interaction may be considered either from the point of view of competition (supply of the same product, tenders), either from the point of view of cooperation (common resources, mergers and acquisitions of development companies). Besides, there are “vertical” hierarchical relations which have big importance. Those are relations

between developers and banks, administrative agencies, different suppliers (consulting, project, building and other companies). A problem of control of the sustainable development of the RED-activity has a special interest.

It is rational to use mathematical models for investigation and control of the RED-activity. There are many publications on project management including mathematical models and methods (Goodpasture, 2004; Oberlender, 2000; Walker, 2002).

This paper is dedicated to the system of optimization and game theoretic models of the RED-activity; its structure is shown in Fig.1.

Figurel. Hierarchical system of mathematical models of the RED-activity

The basic role in the proposed system is played by aggregated models of a REDcompany. First, these are static optimization models which are aimed to determine optimal prices of the real estate development considering constraints on solvent demand and credit return. Second, these are dynamic models of search of the optimal ratio between sales and rentals of a RED-object.

There are two natural directions of generalization of the basic models: “horizontally” and “vertically”. First, it is possible to consider interaction between developers as equal economic agents. In turn, two modeling variants are possible in this case. If we consider competitive relations of RED-companies without formation of coalitions then non-cooperative games of n players in normal form arise. If a cooperation is admissible (common resources, mergers and acquisitions of REDcompanies) then we get cooperative games. Second, RED-companies have economic relations with organizations of other types. These relations are hierarchical as a rule, and a RED-company can be both a Leader (in relations with its suppliers) and a Follower (in relations with its investors, credit institutions, administrative agencies). Respectively, hierarchical game theoretic models arise.

2. Optimization Models

A static optimization model of finding optimal sales price with constraints on nonsatisfied solvent demand has a form

where j is a RED-project index (combination of the RED type and class); N is a quantity of RED-projects implementing by the developer in the current year; u is an annual profit of the developer ($); Sj is an annual building volume on the j-th RED-project (m2); Cj is a cost price in the j-th RED-project ($/m2); pj is a sales price in the j-th RED-project ($/m2); aj(pj) is a share of the sold m2 of the total amount Sj ; are constant expenditures ($); Smax is a maximal solvent demand of the developer target consumer group (m2); pmax is a maximal possible sales price in the j-th RED-project ($/m2).

Let’s accept the following considerations:

- variables which don’t depend on p may be excepted from the payoff function;

- it is convenient to describe a solvent demand by the parameter ¡3 = Smax/Sj,

where all variables relate to a separate RED-project.

Models (1)-(2) or (3)-(4) are static ones, i.e. they describe an activity of a REDcompany during a year. The key role in the model (3)-(4) is played by the function a(p) which describe the dependence of sales share from sales prices. A parameterization of the function a(p) is based on the following proposals which don’t constraint a generality:

- a(p) is a decreasing function of sales price, 0 < a(p) < 1;

- a(o) = 1, a(pmax) = 0.

The simplest function which satisfies the constraints is a linear function

N

o < 3 < 1;

- index j may be omitted without lost of generality. Then we get

u = a(p)p ^ max

(3)

a(p) < ß, 0 < ß < 1, 0 < p < pmax,

(4)

a(p) = 1 - p/pmax;

(5)

The solution of the problem (3)-(4) with (5) gets

and

So, if ¡3 decreases from 1/2 to 0 then the optimal sales price p* increases from pmax/2 to pmax but the profit u(p*) nevertheless decreases from pmax/4 to 0.

Here U is a total developer’s profit ($/m2); T implementation period (months); s is sales price ($/m2); c is cost price ($/m2);

r is rentals rate for a month ($/m2); z is maintenance cost for a month ($/m2); K1(s,c) is profit from sales considering taxes ($/m2);

K2(r, z) is profit from rentals considering taxes ($/m2); f3t is the total share of m2 sold and rented in the month t; at is the share of sold m2 in the month t.

Considering that the payoff function is linear on the controlled variable at we get the optimal solution as

For example, if the whole RED-object is sold or rented in the first month then rentals are more profitable than sales if K1(s,c) < TK2(r,z), and if the object is sold and rented uniformly during the whole period T then the condition becomes

3. Games in Normal Form

Now let’s consider several RED-companies acting on a territory. Let’s designate them by an index i = 1,...,n. Then a competitive interaction of the REDcompanies is described by a n-persons game in normal form

where payoff functions ui are given by the formula (1) and sets of admissible strategies Xi defined by the constraints of the type (2). The following hypotheses about the game theoretic model (7) were investigated:

1) a = ai(pi), 0 < Pi < pmax, i =1,...,n,

where p'nax is the maximal admissible sales price, i.e. ai(p'nax) = 0;

2) ai = ai(priel), priel = Pi/Pmax, Pmax = max{pi,... ,pn};

3) Xi is determined by constraints aiSi = Smax for each RED-company i =

1 ,...,n independently;

4) Xi is determined by common constraints aiSi = Smax for the total solvent demand of the population of the territory.

In all four cases of possible combinations ai and Xi the character of solution (6) is qualitatively the same.

The dynamical model of search of the optimal ratio between sales and rentals in commercial RED-projects has a form

T

T

U = K1(s,c)J2 atßt + K2(t,z) Yp -1+i)(i — at)ßt ^ max

t=i

t=i

T

J2ßt < 1, ßt > 0, 0 < at < 1.

t=1

t=i

0, otherwise.

1, Ki(s,c)E ßt >k2(t,z)Y,(t — t + 1)ßt,

T

t=i

T

Ki(s, c) < 0.5(T + 1)K2(r, z).

G < {\,...,n'}, {^l,...,Xn}, \_ui,...-1 un\ >,

(7)

Optimization and Game Theoretic Modeling of the Real Estate Development 307 As far as the solution (6) is a dominant strategy of the player i then the vector

p* = (p1 ,...,p*n) (8)

may be considered as an equilibrium in dominant strategies in the game (7). But it is important that the players behavior is absolutely independent almost in the case ai = ai(pi) , aiSi < Snax . In another three cases the calculation of dominant strategy requires from a player the knowledge of another players parameters, hence the solution (8) must be more correctly treated as Nash equilibrium which allows some informational exchange between players.

4. Cooperative Games

Let’s consider again several RED-companies acting on a territory and designate them by an index i = 1,...,n. Let’s suppose that the companies can exchange information, join the resources and implement common projects. Let Ai be the amount of actives of the i-th RDE-company.

Then cooperative interaction of the companies may be formalized as a voting game (Amin; A1,..., An) , i.e. the characteristic function has a form

( 1, E Ai > Amin, v(S) = \ ies (9)

0, otherwise.

So a coalition is winning if and only if for this coalition E Ai > Amin. The threshold value Amin may be treated for example as a deposit necessary for participation in a tender or for providing a bank credit.

It is possible to mark the following specific cases of the game (9):

1) dictator game 3 i G {1,...,n} : Ai > Amin,yj = i Aj < Amin . In

this case the game is non-essential, v(S) = 1 i G S, and the only imputation (0,..., 0,1,0,..., 0)(xi = 1) exists which forms -core, is the only NeumannMorgenstern solution and the Shapley value;

2) symmetrical game of the k-th order

v(S) = {0, otherwise. s = |S1,1 < k < n.

In this case -core of the game is empty, the Shapley value has a form

(1/n,..., 1/n), an example of the Neumann-Morgenstern solution is the set

{(xii,..., xHk, 0,..., 0) : xii > 0,...,xHk > 0; xii + .. + xHk = 1}.

It is possible to consider cooperative games in general form where a coalition formation S T means merger (acquisition) of RDE-companies S and T or simply joining of their resources.

5. Hierarchical Games

Let’s consider as an example the interaction between RED-companies and a bank (supposing for simplicity that the only bank credits developers on the territory). The game is based on the following rules.

Stage 1: preparation of credit applications by RED-companies.

This stage includes for each RED-company i = formation of concepts of

the RED projects j = 1,...,ni ; working out schedules of projecting, construction, and financing for each project; estimating of the actives and cost prices; investigating the needs in credits and applying the bank for credit

Ui

= £ K°,.

j=i

Stage 2: decision making by the bank. In this stage the bank analyzes credit applications K0,..., KU; estimates credit risks vi for each application; determines credit rates si = si(ri); makes a decision on credit payments K1,..., KU and corresponding credit rates si,..., sU; informs RED-companies about the decision.

Stage 3: decision making by developers. In this stage each RED-company

i = 1,...,n: corrects real volumes of construction works and corresponding schedules based on credit amount Ki and credit rate si; determines the optimal sales price by solving an optimization problem (3)-(4).

The model of bank decision making is based on the following simplifying hypotheses:

- credit risks are estimated as

Vi = Ki/Ai, i =1,...,n, (10)

where Ai are actives of the i-th developer, Ki are credit amounts. Then the condition of crediting is given by inequality vi < rmax , where rmax is a normative of acceptable risk for the bank;

- credit rate is an increasing linear function of the risk:

si = ari + b = aKi/Ai + b = aiKi + b, i = 1,...,n. Let’s consider that

0 ^ smin < si < smax ^ 1,Vmin ^ Vi ^ Vmax, s(vmin) = Smim s(vmax) = smax-

Then we get

ai = (smax smin)/[Ai(rmax Vmin)], b = (sminVmax smaxVmin)/(vmax Vmin),

i = 1,...,n.

Considering the accepted propositions the model of bank decision making in stage 2 is an optimization problem

UU

u0 = ^ SiKi = Yj(aiKi + b)Ki ^ max (11)

i=1 i=1

U

YjKi = K, 0 < Ki < Li, i = 1,..., n, (12)

i=1

where is a total capital of the bank in the current year, Li = min{Ki0,Airmax}. Solving the problem (11)-(12) by Lagrange method we find optimal values

K* = min{Li, Mi}, Mi = K/(aiY, a-1); (13)

Si = [(smax Smin)Ki + Ai(sminVmax SmaxVmin )] / [Ai (vmax Vmin)],

i = 1,...,n. (14)

The model of developer decision making in stage 3 has a form (3)-(4) with additional constraint

and the corresponding value ß* = S™ax/S*, which has to be substitute in the formula (6) to calculate the optimal sales price.

Let’s consider the case of interaction of the only RED-company with the bank. Then we get a hierarchical game “Bank-Developer” in the following form:

The outcome (Ki,p1), where K* is calculated by formula (13), and pi by the formula (6) after substituting the values s* and S* by formulas (14) and (15) respectively, is the Stackelberg equilibrium in the game (16) - (19).

The problem of control of the sustainable development (Ougolnitsky, 2002) in the case of RED may be formulated as follows. A tree-like control system is considered where the upper level is presented by Administration, and the lower level by Developers designated by index i = 1,...,n. Each Developer maximizes its profit (probably with constraints on the solvent demand). Administration solves two problems. First, it tends to develop the regional construction works complex, or in model terms to maximize the summary profit of Developers with consideration of control expenditures. Second, it has to provide sustainable development conditions which mean in model terms some necessary constraints on social class residential RED volumes.

In general model of hierarchical control of the sustainable development (Ougol-nitsky, 2002) the Principal can use control methods of compulsion (administrative impact), impulsion (economic impact), and conviction (psychological impact). In the described model of hierarchical control of the sustainable development of the regional construction works complex Administration can’t use compulsion because it has no legal possibilities to compel Developers to build social class houses. In return, it has many economic possibilities of impulsion: warranties of buying of social class apartments on the previously stated price, state warranties for bank credits, direct grants and so on. Theoretical possibility of conviction (voluntary cooperation of Developers with Administration) also exists. The model of hierarchical control of the sustainable development of the regional construction works complex may be presented in the following form:

(15)

uo(Ki) = aiK^ + bKi ^ max 0 < Ki < min{K, K0, Airmax} ui(Ki,pi) = [a i(pi)pi - ci][Ai - Ci + (1 - s i)Ki]/ci ^ max 0 < a i (p i )[A i — Ci + (1 — s i )Ki]/c i < 0 < p i < p^ax.

(16)

(17)

(18) (19)

n

uo(p, S) = Y Ui(pi,Si) — fo(p) ^ max,

(20)

i= i

pi e Pi, i =1,...,n;

(21)

Ysn > Smin; (22)

i=1

Ui(pi,Si) ^ max, (23)

Si e £2i, i = 1,...,n. (24)

By the index j = 1 are designated social class RED projects; Sij are volumes of construction works for the j-h project for the i-th Developer; S™ is the necessary volume of social class construction works, i.e. inequality (22) reflects social requirements to the sustainable development of the regional construction works complex; Si = (Si1,... ,Sini), where ni - is total quantity of projects implemented by the

i-th Developer; S = (S1,..., Sn) ; = (p1,... ,pn) is a vector of impulsion controls

used by Administration; fo(p) is control expenditures function of Administration; ui is payoff function of the i-th Developer; Qi - set of constraints for the i-th Developer. In this model Developer’s strategies are not sales prices but construction works amounts; sales prices are determined by the combination of type and class of the RED project.

An outcome

(pl,... ,p*n, ...,Sl,...,S*n) e Pi X ... X Pn X Hi X ...X fin is named the solution of the hierarchical game (20)-(24) if

uo(p1,... ,p*n, ...,S1 ,...,S*n) = max min uo(pi,... ,pTi, ...,Si,..., Sn),

PiePi SieRi(pi),i=1,...,n

where Ri (pi ) {Si e Hi . ui (pi, Si) max ui (pi , zi)}, i 1,...,n,

ZiEOi

with the obligatory condition (22).

Now let’s consider a pricing model of hierarchical control of the sustainable development of the regional construction works complex. It has a form

ul(p,po,p) = Spa(p) - Mp(p,po) ^ max (25)

0 <po < p < pm ax

(26)

uf(p,po,p) = pa(p) +p£(p)(1 - a(p)) ^ max (27)

0 < p < pmax (28)

Here p is sales price; po - normative price of social class residential real estate development; p - limit price of social class residential real estate development;

M >> 1 - penalty constant; p(p,po) = 11,^> > p° ; 5- Administration bonus parameter for social class residential real estate development sales; pmax - ”overlimit” price of social class residential real estate development (there are no sales if p > pmax); a(p) is share of residential real estate development bought by Administration with warranty; £(p) - share of another residential real estate development successfully sold by Developer without help.

Variable p is Developer’s strategy, and variables p0 and p are strategies of Administration which tries to satisfy the sustainable development condition

P < po. (29)

Functions a(p) and £(p) decrease on the segment [0,1] and satisfy the propositions a(p) = < 10 “ P _ Po , £(0) = 1, £(p) = 0,p > pmax. Let’s consider as

[ 0,p > po

illustration linear functions a(p), £(p), then we get 0,p > p;

Up =

P, 0 < p < po,

pla^(.p-p0)p+p0p2-p3 r)n < r> < r Pmax (p PO ) >P0<P<P,

P - <P <P

max?

. 0,p > Pm

Hence we have:

- when 0 < p < p0 then the maximal value uF achieved by p = p0 and is equal

top0; _______________

- when Po < P < P = 0 if P = Po±VPo+4P™^(P P°). one roots is

negative but the positive root is more than p, hence the maximal value uF achieved

-2

by p = p and is equal to p — —;

- when p < p < pmax the condition = 0 gives p* = If p <

then the maximal value up achieved by p = p* and is equal to otherwise

—2

(p > £sp^) it achieved by p = p and is equal to p — ^—; let’s gather the found values in a table.

Values of [0,Po] [po,p] \p, pmax]

p < p >

P* Po P Pmax P

Up Po F Pmax ’ Pmax 4 F Pmax ’

Note the following:

- closed ranges (segments) of p values may be used, because values of uF on the ends are equal on each side;

-2p

- for each p is true p — ;

- if limit price of social class residential real estate development is high (p>£i^i)

then the optimal price is bigger than if it is low (pK^2^), and the Developer payoff is less in this case. But it is dangerous for Administration to establish lower limit price because in this case to achieve the maximal payoff u*F = Developer uses the strategy p* = which may force the sustainable development condition

(29).

Administration can provide the sustainable development by the control mechanism po > , p > . Then the strategy p = p is not profitable for Developer because u(p) < and the strategy p = is not admissible. That’s

why Developer has the only optimal response p = p0 , which satisfies (29), and up = ml = . The outcome + e, is a conviction equilib-

rium.

6. Conclusion

The problem domain of real estate development is conceptualized in terms of optimization and game theoretic models. A system of such models is proposed. The system is based on optimization models of separate RED-companies which are generalized as games in normal form, cooperative games, and hierarchical games. The problem of hierarchical control of sustainable development is also adapted for the RED problem domain. Some examples are driven.

References

Goodpasture, J. C. (2004). Quantitative methods in project management, Roca Baton, Ross Publ.

Oberlender, G. D. (2000). Project management for engineering and construction, N.Y., McGraw Hill.

Ougolnitsky, G. A. (2002). Game theoretic modeling of the hierarchical control of sustainable development, Game Theory and Applications, 8, 82-91.

Walker, A. (2002). Project management in construction, CityplaceOxford, Blackwell Science.