United States 

Department of 

Agriculture 

Forest Service 

THEORY OF GAMES 

AND APPLICATIONS 

N FORESTRY 

Southeastern Forest 

Experiment Stat ion 

Benee F. Swindel 

General Technical 

Report SE-26 
May 1984 

Southeastern Forest Experiment Station 

Asheville, North Carolina 
Contents 

Page 

1. Introduction and Theoretical 

Aspects 

Game Theory in Forestry ................ 1 

Zero.Sum. Two-Person Games .............. 3 

Strictly Wtermined Games ............... 3 

Inadmissible 9tra tegies ................ 4 

Computations for Strictly 

DeterminedGames .................. 4 

Mixed Strategies ................... 4 

ExpectedPayoff .................... 5 

The Fundarnentax Theorern ................ 6 

b1ention of Other Games ................ 6 

Decision Theory ................... 7 

11 . Procedural Aspects and 

Illustrative Applications 

I1 . 1 Gocations for Two Companies 

ioaForest .................... 7 

11.2 When to Patr#~l ................... 9 

11.3 A Graphic31 Met\od for 

Finding an Optimal Strdtecjy 

for Any Player With Two 

Strdteyies ..................... 10 

I11 . Programing for Computer 

Execution ....................... 12 

IV. Suggested Reading 

TV.1 Elementary Texts .................. IV . 2 Oti-ler Texts ..................... IV.3 Other References ................... IV.4 Ilecision Theory in Forestry .............. 

Theory of Games and Applications 

in Forestry 

Bence F, Swindel, Project Leader 

Intensive ihnagement Practices 

Assessment Center 

Gaincsville, Florida 

I, Introduction and Theoretical Aspects 

1.0, Game Theory in Forestry 

The theory of gdmes of strategy derives from von Neumann and kjorgenstern 

( 1944) , Their book was greeted with 

considerable enthusiasm by military 

s tuategis ts , economis ts , and others. 

The elegance of their development of the 

theory is much admired, Still, substantive applications within forestry and 

other natural resources fields have been 

rare. The root cduse of this paucity of 

applications is not clear, Surely the 

theory is little known to forest managers and decisionmakers. All this 

suggests that an exposition of game 

theory for natural resources managers 

could encourage application of the tcchniques in solving real problems. Yence 

this paper. Aside from that, the theory 

has an intrinsic beauty and study of 

such ahs tractions, as of mathematics 

generally, inculcates lof ty hahi ts of 

mind, 

There are very obvious limitations to 

the applical2i li ty of game theory, Thus, 

two-person, cons tant-sun games assume 

two knowledgeable and selfish adversaries with opposed interests. Clearly, 

the adversaries can he two people, but 

need not be. They can as well be two 

agencies, two companies, two coalitions, 

4x0 armies. All that is required is 

that individuals in each group share, so 

far as the game being played is eoncerned, a common interest , 

The interests must be strictly opposed. 

One player's gain is the other's loss. 

Each is assumed knowledgeable of all 

options and their consequences. Each is 

expected to try to maximize his own 

gain, implying he seeks to maximize his 

opponent's loss, Such situations must 

be rare within a single organization 

wit11 a united purpose. Thus, within a 

cornpany or agency managing land for 

profit or public good, it appears that 

opportunities for strict applicability 

of this theory as a guide to rational 

behavior are indeed rare, Even between 

companies within an industry or be tween 

departments within an agency, compelling 

examples of complete antagoriissn do not 

readily come to mind in profuse 

abundance, 

Still, there are instances of conf licting situations that can be described 

fundarnentally as a game of strategy between two or lnore adversaries, The 

insight provided by the game theoretic 

formulation and solution can be a helpful guide to choosing, predicting, or 

understanding rational behavior in such 

circumstances . A few examples that may 

be suggestive of many are given in 511. 

Those are necessarily much simplified in 

order to treat them in the space allotted and so that the technicpes of problem solving may not be lost in the 

complexities of more realis tic 

situations. 

In order that the theory may not seem 

trival, we now attempt to show that the 

logic of game theoretic abstractions has 

profound implications for large problems 

(as sell as delightful inplications for 

small ones). To do so we extract freely 

from Garrett Hardin (1 968) , Hardin's 

thesis was that the world population 

problem has no technical solution hut 

requires rather a fundamental extension 

of morality. Some of his analyses are 

used out of that context--since they 

apply equally to the one we consider. 

With regard tc:, some of its attributes, 

the forests of a nation are often conceived as a commons, Their beauty and 

protective inf iuence over water and 

wildlife resources are thought of as a 

natural resource, Coricerned citizens 

frequently express an interest in the 

conservation of these resources and 

sometimes object to mnagement measures 

implemented to achieve the immediate 

objectives of the landowner. 

In Hardin ' s words : 

"The tragedy of the commons develops 

in this way, Picture a pasture open to 

all. It is to be expected thdt each 

herdsman will try to keep as many cattle 
as possible on the commons. Such an 

arrangement my work reasonably satisfactorily for centuries because tribal 

wars, gmaching, and disease keep the 

numbers of 'mth man and beast %sell helow 

the carrying capacity of the land, 

Finally, however, comes the day of 

reckonf ng, thdt is, the day when the 

long-desired goal of social stability 

becorner; a reality. At this point, the 

inherent logic of the coinmons remorselessly generates tragedy. 

"As a rational being, each herdsman 

seeks to raaximize his gain, Explicitly 

Dr implicitly, more or Less consciously, 

he asks, 'What is the atility to me of 

adding one ;ixore animal tf) my herd?' 

This utility has one negative and one 

pos i tive component. 

( 1 f The pcsitive component is a 

function of the increment of one anitrtal . Since the herdsman receives all the 

proceecJls from the sale of the ctdditional 

animal, the positive utility is nearly +I, 

(2) The negative component is a 

function of the additional overgrazing 

created by one more animal. Sf nce, however, the effects of overgrazing are 

shared by all the herdsmen, the negative 

utility of any particular decision-making 

herdsman is only a fraction of -1. 

"Adding together the component partial 

rztilities, the rational herdsman 

concludes that the only sensible coarse 

for him to pursue is to add another 

animal to his herd. And another; and 

another.... Rut this is the conclusion 

reached by each and every rational 

herdsman sharing a cornmons. Therein is 

the tragedy. Each man is locked into a 

system that compels him to increase his 

herd without limit--in a world that is 

limited. Ruin is the destination toward 

which all men rush, each pursuing his 

own best interest in a society that 

believes in the freedom of the cornmons, 

Freedorn in a cornmons brings r~in to 

all Ii 

Do these concepts of a constant-sum 

game, and the tragic implications of 

dominated strategies so graphically 

described, have any analogous implications in the colnplex problems of formulation of forest resources management 

policy? Hardin himself gives hoth a 

specific and general illustration ::hat 

establishes the affirmative. 

"The National Parks present another 

instance of the working out of the tragedy of the commons, At present, they 

are open to all, without Limit. The 

parks Elernselves are limited in extent-- 

there is only one Yosemite Valley-- 

whereas population seems to grow without 

limit, The val~les thctt visitors seek in 

the parks are steadily eroded. Plainly, 

we must soon cease to treat tile parks as 

commons or they will he of no value to 

anyone. 

"What shall we do? We have several 

options* We might sell them off as private property. We night keep them as 

public property, but allocate the right 

to enter tllem. The allocation might he 

on the basis of wealth, by the ase of an 

auction system. It night he on the 

basis of merit, as defined by some 

agreed-upon standards. It might be by 

lottery, Or it rnight he on a first-come, 

firs t-served basis, administered to long 

queues. These, T think, are all the 

reasonable possibilities. They are all 

objectionahle. Rut we rctus t choose--or 

acquiesce in the destraction of the cornmons that we call our 'IJational Parks.. . In a rzverse way, the tragedy of the 

commons reappears in problems of pollution, Here it is not a question of 

tdking something out of the commons, but 

of putting something in--sewage, or chemical, radioactive, and heat wastes into 

water; noxious and dangerous fumes into 

the air; and distracting and unpleasant 

advertising signs into the line of 

sight. The calculations of utility are 

much the same as before, The rational 

man finds that his share of the cost of 

the wastes he discharges into the commons is less than the cost of purifying 

his wastes before releasing them, Since 

this is true for everyone, we are locked 

into a system of 'fouli~g our own nest,' 

so long as we behave only as independent, rational, free-enterprisers ." 

Hardin concludes that the solution to 

producing temperance in the use of a 

corninons resides in mutual coercion 

mutually agreed upon. Importantly, he 

re~narks : 

"It is worth notinq that the mortality 

of an act cannot be determined from a 

photograph. One does not know whether a 

man killing an elephant or setting fire 
to the grassland is haraing others until 

one knows the total system in which his 

act appears, 'One picture is north d 

thousand words,' said an ancient Chinese; 

but it may take 10,000 words to validate 

it, It is as tempting to ecologists as 

it is to reformers in general tr> try to 

persuade others by way of the photographic shortcut, But the essence of an 

argurnent cdnnot- be photographed: it 

must be presented rationally--in words," 

The compelling logic of the foregoing 

analysis leads to the coneLilsion that 

policies to protect forest resources 

will best rely on mutual coercion 

miltdally coneei ved from reliable evi - 

dence concerning the f l~nctioni ng of the 

total forest ecosys tem, Agai~ quoting 

Hardin, 

"reaching an acceptable and stable 

solution will surely require more than 

one generation of hard analytical work-- 

and much persuasion," 

Persuasion and agreement depend on unders tanding the analytical work--including 

the tools of the analysis. OccasionaLly 

the appropriate dnalytical tool is the 

theory of games of strategy. 

1.1. Zero-Sum, Two-Person Games 

The following abstraction taken as a 

definition of a game of strategy will he 

suf Eiciently general Fi-)r ws t purposes, 

One of two players is required to choose 

one among the finite set of strategies 

The second player is required to choose 

one strategy frola the finite set 

Each player is aware of all of his own 

and a11 of his opponent's alternatives, 

but mast choose his own strategy without 

benefit of knowing his opponent's 

choice, Subsequent to a choice of strategy, say ai, by Player I, and a choice 

of strategy, say hj , by Player 11, 

Player I1 is required to pay Player I an 

amount ei (~ossibly negative ) . Such 

games are called two-person games for 

obvious reasons and zero-sum games since 

the sum of the winnings of the two 

players is zero; there is no "house," 

Clearly, zero-sum, two-person games are 

characterized by their payoff matrix 

which is known to both players, 

Player I1 Strategies 

bl b2 eee bj erne bn 

It is obvious that Player I1 vrould 

like to choose a strategy so that the 

payoff (which is his loss) is as small 

as possible, But, the payoff depends on 

the strategy chosen by Player I. So, the 

question arises as to whether Player I1 

has an optimum strategy. And conversely 

for Player I, who wishes to make the 

payoff (which is his gain) as large as 

possible, 

I. 2. Strictly Determined Games 

In the payoff matrix for certain games 

there is a payoff, eij say, simultaneously not larger than any payoff in the 

same row, and not smaller than any payoff in the same column, Such payoff s 

are said to 'oe a saddlepoint, the game 

is said to be strictly determined, and 

gi j is called the value of the game . Consider the implications to the two 

players should the payoff matrix in $1.1 

contain a saddlepoint at Xije ~f Player 

I then chooses the strdtegy ai, he guarantees himself a payoff (gain) of at 

least gi j --no matter what strategy 

Player 11 chooses. Conversely, if 

Player I1 chooses the strategy bj, he 

guarantees himself a payoff (loss) of no 

more than gi j . In this situation, it is 

clear that neither player may, by any 

alternate choice of strategy, hope to 

obtain more favorable results, Player I 

cannot do better than choose ai, which 

is called a maximim pure strategy and is 
optimum for him, Player I1 cannot do 

better than choose hj, which is called a 

minimax pure strategy and is optimum for 

him. The payoff is then Rij = v, say 

the value of the game. 

I,3, Inadmissible Strategies 

In some payoff matrices, a systematic 

search for optimum strategies is facilitated by deleting certain rows and/or 

columns. In the payoff matrix of $1~1, 

suppose the first t~o columns possess 

the property: 

'il gi2, a11 i = 1, 2, .. ., m, and 

Ril > Rial some i = 1, 2, ..,, m, 

i .eel in each row Lhe payoff in column 1 

is always as large as the payoff in 

column 2--and sometimes larger, In that 

cdse, choosing b2 is always as good as 

choosing hl for Player 11, and sometimes 

(for countering some choices of his 

opponent) it is better. Player I1 can 

have no redson ever to prefer strategy 

hl to b2, and hl is said to be dorninated by b2. Any strategy of either 

player so dorninatt?d by an alternative 

strategy is said to be inadmissil3le 

and may be deleted from further 

consideration, 

1,4, Computations for Strictly Determined Games 

The payoff matrix of $1~1 may be 

systematically examined to determine 

whether it has a saddlepoint or not hy 

appending a column (coinposed of the row 

minima) and a row (composed of the 

column maxima) as follows, 

1 Player TI Strateqies I 

w 

ii min R a a 521 gZ2 ... 11 2n k j zj 

If the maximurn of Lie elements in the 

appended column (i ,e ,, the maximum of 

the row minima) is equal to the minimum 

of the appended row (i ,e,, the minimum 

of the colurnn maxima), then the game is 

strictly determined (i.e., the payoff 

matrix has a saddlepoint), Moreover, 

any strategy of Player I that xaxi~rtizes 

the row ininima is a maximin strategy and 

an optimun strategy for him, Conversely, any strategy of Player I1 tnat 

minimizes the column maxima is a minimax 

strategy and an optimun strategy for 

him, Finally, the value of t'rle game is 

v = max min R = min max R , i j i j i j j i 

It may be noted that a payoff matrix 

may have more than one saddlepoint, The 

payoff is always v at each of tilern. And, 

when either player has more than one 

optimum strategy, he will he indifferent 

hetueen them since each guarantees a 

payoff of v, 

I ,5, Hixed Strategies 

We have completed our theore tical 

description of strictly determined games 

of strategy, Henceforth we are particularly interested in garnes with 

payoff matrices where 

max min R < min max R i j i j i j j i 
(the opposite inequality is arithmetically impossihle). From $1.4 it is 

obvious that Player I1 always has a pure 

strategy khat ensures his Loss will he 

no greater than the right-hand side of 

the last inequality, Analogous1 y, 

Player I has a pure strategy timt 

ensures his gain will be no less than 

tlie left-hand side, In $1.4 these two 

yua:ltities were equal, and discovery of 

such strategies constituted a solution 

of the game, 

Now the disparity between the two 

sides of the last inequality may be 

viewed ol3portunis tically by both 

players--f or it suggests the possibility 

of increased gain (reduced loss) over 

that guaranteed by the maximin (minimax) 

pure strategy, The ftlnda~nental theorem 

of games addresses this possibility and 

provides a definitive solution for each 

player, It employs the concepts of 

mixed strategies for the players and 

expected payoff in the repetitive play 

of the same game. 

Consider then that Player I can elect 

to choose among his pure strategies, 

a1 , with the aid of a random device, 

Thus, with such a device, he might 

choose strategy a1 with probability 

pl; a2 with probability p2, .,., 

and strategy a, with probability p, 

where: 

m . O< pi, i = I, 2, , m; and 1 = C Pi 

i=l 

(For example, if in each play of a certain garne, Player I has only two pure 

s trategies, al and a2, and he chooses 

a1 if the flipping of a (fair) coin 

shows heads, and he chooses a2 if the 

coin shows tails, then m = 2 and p1 = 

P2 = l/2* Choice of a particular set of 

probabilities 

constitues choice of a mixed strategy by 

Player I, The set of all mixed strategies available to Player I is the set of 

all I xm vectors, such as - p above, where 

the elements of - p are nonnegative and 

sum to one , 

Analogously, Player I1 could choose 

among his pure strategies, bl , b2, -, 

bn, with probabilities 

i ,e,, he could choose a mixed strategy 

by electing one among all lxn vectors 

whose eleaents are nonnegative and sum 

to one, 

I, 6. Expected Payoff 

Given a particular choice of a mixed 

strategy, - p, by Player I, and a partitular choice of a mixed strategy, q, 

by Player 11, it is a simple computation 

to find the expected payoff in a long 

series of repetitions of a zero-sum, 

two-person game, For the expected 

payoff is the sum over all prospective 

payoffs of the product of the payoff and 

its probability of occurrence, Clearly 

the probability of dny payoff in the 

payoff matrix, say Xijl is 

since kij is realized if and only if 

Player I chooses his strategy ai (which 

he does with probabi lity pi ) , and Player 

I1 chooses his strategy bj (which he 

does with probability q, ) , and since 

($1.1) choice by each player is made in 

the dhsenee of knowledge, and therefore 

independently, of his opponent's choice. 

Thus, given 2 and - q, the expected payoff 

is 

where - q4n,l is the matrix transpose of 

%lxnr L = &,, is the payoff matrix, and 

2 L q, is simply the matrix product 

which is a scalar, i.e., one number. 
I,7, The mdamental Theorem 

The fundamental theorern of games of 

strategy given by John von Neuaann 

established that for every payoff matrix 

L 

max min - p L q* 

- p s 

and 

min max - p L 3" 

- s P 

both exist and they are equal. Their 

cornmon value, say V, is called the value 

of the game, Stated another uay, the 

theorern establishes that in any zerr>- 

sum, two-person game Player I has a 

strategy, p*, among his strategies p, - and player-11 has a strategy, - q*, among 

his strategies q, such that - 

Thus, frorn the left inequality, Player 

I1 has a strategy, q*, that cjuarantees 

that his expected loss will not exceed V 

no matter what strategy Player I 

chooses. And, f rorn the right inequality, 

Player I has a strategy, p*, that 

guarantees that his expected gain will 

be at least V no matter what strategy 

Player I1 chooses. Since neither 

player, by alternative choice of strategy, can improve his prospects, each may 

as well choose p* and q*, respectively, 

so as to achievg V in The long run: - p* 

is a maximin mixed strategy and optimun 

for Player I; q* is a minimax mixed 

strategy and optimum for Player 11. 

It should he observed that the choice 

of a pure strategy, ail by Player I, 

i .eel the certain choice of ai can he 

described as the choice of the mixed 

strategy with pi = 1 and pic = 0, where 

k f i. And, similarly, the choice of 

the pure strategy, bj , by Player I1 can 

be described as the choice of the mixed 

strategy where all elements are zero 

except for one in the jth position. 

When the payoff inatrix has a saddlepoint, the maximin pure strategy satisf ies the defining property of p*, the 

optimun mixed strategy for player I, and 

the minimax pure strategy satisfies the 

defining property of - y*, the optimum 

mixeci strategy for Player 11. 

I,8, Pilention of Other Games 

Extensions in several (liree tions of 

the theory of tile preceding sections 

have been attempted. Some of them will 

now be briefly reviewed. 

Games involving three persons, four 

persons, and in general many persons 

have been formally considered. Yere 

we will confine ourselves to quoting a 

single paragraph frorn Dorfrnan and others 

(1958): 

"The theory of inany-person qarnes in 

the hands of von Neumann and Moryenstern 

is essentially a theory of coalitions, 

their formation and revision. The 

underlying idea is that two persons in 

such a situation cannot do worse by 

acting jointly than by acting severally, 

and nay do better. Thus a many-person 

game tends to he reduced to a 

two-'person 'game in which each 'person ' 

is a coalition. The problems then 

become: which coalitions will form and 

how will the winnings be divided among 

the meinbers of the coalition? To pursue 

the answers proposed for these questions 

would lead us into a specialized 

discussion, and since these answers are 

not very satisfactory we ref rain. " 

Extending the results of zero-sum 

games to constant-sum games is easy. 

Indeed, in some elementary expositions 

the basic theory is presented on the 

basis of the assumption that the sum of 

the winnings of the two players is 

always a constant, say Re We have taken 

R to be zero for convenience, 

Further extension to non-cons tan t-sum 

games is much more difficult. When one 

player's gain is not necessarily the 

other's loss, there is the possibility 

of increased gain by both players 

through cc3llusion and cooperation, 

Unfortunately, as McKinsey (1952) 

observes : 

"Despite the great importdnce of 

general games for the social sciences, 

there is not available so far any treatment of such games which can be regarded 

as even reasonably satisfactory." 
1.9. Decision Theory 

One application of the theory of 

zero-sum, tl~o-person games envisions a 

game of strategy betageen a decisionmaker 

and "nature," Some inodification of t%e 

preceding arguments is required, and 

there is cotlsiderble controversy, 

Moreorier, ever1 a cursory review of the 

subject would require a trea t;nent cornparable in size to that given I~ere to 

garnes of corlflict bet*dnreen selfish opponents. Still, this important 3uhject 

must be mentioned, 

So, we now consider briefly that the 

abstraction of 51.1 is sometimes used 

to describe the situation of a decisionmaker (Player 11) "in the real world" 

conf rorlterl with choosing arnong decisions 

or actions, bl , b2, " a , bq In the 

natural rzsources field, the sets of 

actions confronting various decisionmakers dre as diverse as one can 

imagine. They mdy deal with business, 

forest management, personnel, silviculture, engineering, and selection, its 

maintenance or deployment of equipment. 

The list goes on, Typically, choice of 

a particular decision corninits one for 

the future, Even if it c30esntt, the 

optimux~ choice of action is typically 

not evident due to uncertainties about 

markets, economic conditions , rlat~lrdl 

events such as weather to be encountered, action or state of biological 

agents, etc, In any event, it is 

possible to envision tile applica'i~ility 

of the materials of $1.1 to some of 

these situatir3ns hy tclking Player I to 

be, for lack of a better descriptor, 

"nature" and the strategies of "nattlre" 

to be the various events or conditions 

thdt the decis ionmaker may conf rant, 

The payoff, Ri j, hecoroes the cos t 

(possibly negative, indicating a net 

benefit) of taking action, hj, should 

condition ai materalize, f learly, 

Player 11 wishes to choose an action, 

hj, to ~ninimize his costs (maximize his 

benefits), Rut, in the formulation of 

$1.1, as in real life, uncertainty 

about the condition, ai_, thdt may prevail creates uncertainty about the 

desirable action, 

NOW, a disparity, Previously, Player 

I was considered to 'be an intelligent, 

deliberative opponent, irlformed of the 

payoff natrix, and expected to choose a 

strategy, air to ntaximize Player 11 s 

loss. "Natare , " ~het%er denoting next 

i~r3nth 's v~eather, tbe biological agents 

at mrk in a forest, or general economic 

coildi tions expected for the next 

quarter, can 'Ilard2y be construed as so 

perverse. StilL, on occdsion, t%e minimax solution given by game theory is 

recommended for decisionnakers who 

simply wish to guard agairlst the v~orst 

contingency. 

More often, an alterrldtive base on 

praynos is is recornmendetl . For example, 

if wedther is the adversary, and potential conditions are enumerated, then 

some thing of the probahili ty of their 

occurrence must he known--perhaps from 

cli~atic records. If the period of tiise 

of concern to the decisionmaker is near 

and. short, perhaps the historical frequency of conditions may be moc"iiied by 

a weather forecast , Analogoasly, enotx~h 

of the health of a forest :nay he known 

to specify the likelihood of vdrious 

biological conditions, and there dre 

f orecas ts of future economi~-: conditions , 

What is usually recornmended to the decisionnaker equipped with information on 

the likelihood of various conditions i.s 

that he deternine his expected loss 

under ench of his alterridtives and 

choose that one minimizing his expected 

loss given his information, That is, a 

Bayesian strategy, is usuall-y recommended in such f~)rrnul~'tti~ns as better 

justified thin d minimax one, 

For tile redder interes ted in decision 

theory, several references in ~IV are 

selected Erorn a voluminous literature, 

11, Procedural Aspects and Illustrative 

Applications 

II,1. Locations for Two Companies in a 

Forest 

Suppose a single railroad traverses a 

homogeneously forested region, Within 
the forest, distances along L3e railway 

from one koun,jary to the other 3re 

scaled From zero to 100. Suppose tao 

fores t pxx'iuc ts companies ir tend t.3 

enter this fnarket by coilstri~er--irtjji a 

single rail sitling enzt~ where d~i~oocl i?dy 

be scd led, 2arclilas z:l, an13 Loa:ried E%J r 

trdnsport to kl~e'i r respective mills. 

They wds t locclts nt tdo poL"~ts, SUCSI as 

1 and It, irl the Ey>llc>wirlg fbgure, 

If de construct 4 point, HI halfday 

beteen I ant3 IT. 3nd assume ea.ch corspany 

will get nll 3f t'w mod ~:~dr'cetet-i From 

it; side c>E poi?t 2, where" shc3aI~;f the 

tdo conzpanies locate their sitlrng? 

This problem f itr; very rleatly into 

the f~rmat of a zero-san, two-person 

game ($1.7 ) if it's assurne:l for 

simplicity th t the companiec; car1 locate 

only at a finite nilmher: of points along 

the railway, say at 0, 20, 40, 50, 60, 

50, and 'I 00, Suppose furt!ler that the 

companies iiriil splits tine market 59-50 if 

they :house the ~~1:rle Location. The payoff .natrix is i;hotlrn in the t.ii2Le helow, 

Ttle 20sses shown in Lhe tlkle are the 

,nrr~purtions <3f t'2e market yielded by 

~Cgil~par~y 11 to Company I, lepending on 

t'?e %do locations chosen. Clearly, 

Gonljany TI wishes to choose a locdtion 

(stsrdtegy) t3 rrti:~-ini.ze stlch a payoff. 

C~rq..any 1 wishes t~ inaxi:n~ze it, 

To rlseerfxio, zrhether this payoff 

matrix hds a sntldlepoint ($1 .I ) , the 

rniqina~rn loss in each row is appended on 

the right of the payoff matrix ($1.4). 

Analogous ly , tile rn(xximt3m loss in each 

column is appendeci to the bott{m, 

Pirrdlly, it is observed that the raaximt~m 

r>E tfle row minima is .50, as is the 

mi?i.nrln of the colil~n maxima, i .em , the 

t*.ro clrr-: 2rl1~al and this payoff mtrix has 

a r;addlepoint, Moreover, the optinurn 

li>crt ti-on (maxirnurn s t ra tegy ) for Company 

f is i:1 the center: (JS the forest (at 

location 50) , 'The optimum location 

(minimax strdlregy) for Company I1 is 

also in the center [at location. 50). 

Locdtion of Company I1 

Row 

20 40 50 60 80 

Column maxima .90 .70 .55 .90 

mini mur:~ 
When both companies locate in the 

center, they share the market 50-50 ( -50 

is the value of the game). The payoff 

corresponding to the two optimum strategies exhibits the characteris tic property of a saddlepoint--it is smallest in 

the row and largest in the columt~ containing it (91.2). 

Notice that lt~cating one company at 

any point other than the center yields 

more than half of the market to the competitor who locates in the center, 

11.2, When to Patrol 

Consider next a problem that confronts, in more or less complex form, 

many fores t ;ttanagers whose johs include 

an element of law enforcement, i ,e,, 

whether to patrol an ared susceptible to 

unLawfu1 entry, Problems of this type 

arise in many forms due to the susceptibility of forest land to timber 

trespass, poaching, arson, etc. 

Suppose a company forester is 

alerted that his lands are the intended 

target of an arsonist. During any particular susceptii>le period ie ,g., 

overnight) the arsonist may either 

attempt to set fire or he my not (i~e., 

he may stdy home), The forester's 

alternative strategies may be to patrol 

or refrain, Suppose the payoff matrix 

is 

Action of 

Don't burn 

The numerical values of these payoffs 

are debatable--as is the ass~~mption of

zero-sun payoff s , Btlt, the rationale 

could be something like this, If 

neither protagonist acts, the (muttlal) 

result is the status quo--indicted by 

zero payoff, If the patrol is employed 

in the absence of the arsonist, the cornpany incurs n cost of 1 man-day of 

unnecessary labor, Alternatively, if 

the arsonist strikes in the absence of a 

patrol, tile company incilrs I0 %an-days ' 

labor in suppressing the fire, 30th of 

these lahor costs are dssumed to accrue 

in satisfaction ti) the arsonist as 

payment for a real or irnagined i-rlj~tstice 

by the cornpany, Finally, if both protagonists nct, the arsonist is cat_tylzt 

and fined and/or iinprisoned at considerable personal loss, The company 

correspondingly gdins from the relief 

from future patrols and suppression frorn 

the arsonist apprehended and perhaps 

from others who might entertain like 

ambitions f~r revenge or mischief in the 

absence of an example of the potential 

consequences, 

This simple game does not possess a 

saddlepoint since 

Inax min ", = 0 < 1 = min max gij, 

i j j i 

i ,e,, the rnaxirn~lr? of the row minima is 

less than the minimurn of tile coluqn 

nlaxima. F-Ience, the optimuxn strategy for 

at least one of the players involves a 

mixture of his pure strategies, and the 

value of this game has not been determined, 

A general algebraic solution for 2x2 

games with payoff matrices 

without saddlepoints is available, In 

such cjarnes, Player 1's optimum mixed 

strategy is to choose his first pure 

strategy with probability p and his 

second wit% probability I -p where 
Analogously, Player 11's optimum mixed 

strategy is to choose his first and 

second pure strategies with probabili - ties q and 1 -g, respectively, where 

Finally, the value of Lhe game is 

{Jsing the numerical payofEs given at 

the beginning of tl?is section and tile 

relevant formulas, it is easy to find 

that 

(i) the c>ptimurn strategy for the 

arsonist is to burn the woods 

with probability p z 1/111 m 

,009 and to stay borne with 

probability 1-p = 110/111 Z 

,991 , 

(ii ) the optimun strategy for the 

forester is to patrol with 

probability q = 10/1'11 Z .09 

and to refrain with prohahility I-q = 101/111"N .91, and 

(iii) the value of the game is 

v = 10/111. 

11.3, A Graphical Method for Finding an 

Optimal Strategy for Any Player 

With mo Strategies 

Without attempting to motivate the 

specific game, we next iZlustrate CI 

graphical technique for finding 

(approxizately) an optiaum nixed strategy for any player who has only two 

pure strategies frorn which to choose, 

Yis opponent lnay have any (finite) 

number of pure s tua tegies , Caref al 

study of this material will facilitate 

understanding of $111, The example here 

is from Singleton and Tyndall ( 1974) , 

Consider the game with payoff matrix 

Player I1 Strategies 
Since the payoff matrix has no saddlepoint, the problem for Player I is to 

find a mixture of his two pure strategies, al and a2, that maximizes his 

minimum expected gain (against any pure 

or mixed strategy of Player 11). Thus, 

Player I seeks a probability, p, with 

which he will choose his strategy 

a1 (and consequently a probability, I-p, 

with which he will choose his strategy 

a,) to rnaximize his minimum expected 

gain. 

Consider that, given p, Player 1's 

expected gain against bl (cf . column one 

of the payoff matrix) is 

This function is plotted and labeled in 

the following graph. It shows, for 

example, that if Player I chooses p = 0, 

so that he chooses with certainty his 

strategy a2, then he gains against 

bl exactly zero (cf . payoff matrix). If 

Player I chooses p = 1 so thdt he 

chooses with certainty his strategy a1 , then he gains against bl one. If Player 

I chooses an intermediate p so that he 

chooses a mixture of al and a2, then his 

expected gain against b1 is given by the 

graph of 5 (p) . Thus, if Player I 

flips a fair coin to choose between 

a1 and "2, his expected gain against 

bl is one-half. 

Similarly, given p, Player Its expected gain against b2, h3, and b4 (cf. 

columns 2, 3, and 4 of the payoff 

matrix) are, respectively, 

c3 (p) = 4p - 3 (1 - p) , and 

All of these functions are depicted in 

the following graph. 
Points on these straight lines depict 

the expected gain of Player I against 

each of the pure strategies of Player I1 

for arty p in the interval zero to one, 

inclusive, iee ., for any mixed strategy 

for Player I. 

Rut, the graph instantly shows more 

than that, For every p, it shows Player 

1's minimum expected gain, *atever 

Player 11's strategy. Player 1's minimum expected gain is 

the hold, segmented linear function in 

the graph. Clearly, Player I maxinizes 

his minimurq expected gain by choosing p 

to maximize 5 (p), ice., by choosing the 

point, ,no, ticked on the graph's axis at 

p = 0.8, Thus, Player Its optinum strategy is to choose 

a1 with probability 0 -8, and 

a2 with probability 0.2. 

Any altarnative choice of 11 clearly 

lowers Player 1's minimum exgected gain 

(cf . graph) , 

then his expected gain against each of 

Player I1 's strategies may be wri tten 

down, These expectations are : 

(against bl) 

(against b 2 ) 

and 

(against bn) 

Now, for the moment, simply define a 

quantity TJ as the minimum of these 

expectations, i ,em, define 

Then (cf, $11~3) Player 1's problem is 

to choose p to maximize V. 

In analogy with problems of linear 

programing, think of Player I as having 

(i) choice variables p .*. 

1' P2f r Pm' and V, and 

111, Programing for Computer Execution 

(ii) the objective of maximizing V. 

Following the strategy implicit in 

$11~3, we can outline a linear 

programing method for the solution of 

general mxn games. 

Remembering the constraints on the pi 

and the definition (immediately above) 

of V, Player 1% problem is to choose 

Plr P2r --* f &% to: 

Consider the game with payoffs 

maximize V such that 

where m and n are (in principle) any 

finite positive integers, Consider 

first the problem of finding an optimum 

strategy for Player I, i .em, an optimum 

vector, p, of probabilities for choosing 

among hiz rn pure strategies al , a2, " , a,. Should Player I choose the particular mixed strategy (cf. $1~5) 
Now, add~ng a constant to all eleaents 

of Lhe payoff rnatrix changes notlliqg as 

far as strategies Edr playing a game are 

eortcerrzed . In piirtlcrllar, cons tan t 

to ix cxddeiJ. may k chosen ti, be -max min 

Eij. This clearly assures t31dt tile LJ 

valize of the gane L:z nor~neyative. Consequently, we may assame that V > 0. 

Then Player 1's problem of findiG a11 

optimurrt strategy is the following linear 

programing program in rimre cotr v\?rl tional 

notation: 

Choose pl , p2, o e sm, V to maximize 

V subject to 

Player XI 9 problem of finding an 

optimum strategy is the Earnorzs duality 

problem of li near 2rugraming. 

IV. Suggested Reading 

IV.1 , Elementary Texts 

NcKinsey, Jo C. C, Introduction to the 

theory of games. New York: 

McGraw-Hill; 1952. 371 p. 

Singleton, R. Re; TyncialL, W, Fo Games 

and programs : !nattlematics for mo,-leling. San Frdncisco: W. R, Freeman; 

1974. 304 p. 

WiLf_i;ins, tJ. 13. The co~npleat strategyst. 

New York: NcGra;~-HFll; 1958. 234 p, 

IV.2, Other Texts 

Dorf-nan, Re; Samuelson, P. A,; Solow, 

Ti. M, Linear programming and econornic 

analysis. New York: McGraw-Hill; 

1958. 524p, 

voil Neu;?lianr~, J. ; Yorgenstern, 0. Theory 

of games and economic behavior. 

Princeton, NJ: '2rinceton Uni \rerui ty 

Press; 1944. 625 12. 

IV.3. Other References 

Hardit?, G, The tragedy of tile cornmans. 

Science 162 ( 3859) : 1243-1 248; 1968, 

Yotalling, ri. Stability in coirlpeti-tion. 

In: Stigler, Go $3.; 'buldi-ny, Ye E., 

eds. %eatlings in price theory, 

Chicdyo: Richacd Do Lrdn; 

1952: 357-484. 

IV.4. Decision Theory in Forestry 

Dane, C. W. Statil;ticcll decision theory 

and it3 application to forest engineering. Journal of Forestry 63 ( 4) : 

276-279; 1965. 

Davis, Jo R. Forest Eire control decision making under conditions of 

uncertdinty. Journal of Forestry 66 : 

626-631 ; 1968. 

Swindel, S. P. The Rayssian eontr~versy. 

Res. p?ip, SE-95. Asheville, WC: 

fJ.S. Department of Agric~lt~lre, 

Fores t Service, Southeas tern Forest 

Fxperilnent Station; 1972. 12 p. 

Thompson, F. F. The theory of decision 

under uncerctiidity aild possil3le 

applications in forest management. 

Forest Science 14: 156-1 63; 1968. 

Jones, A. J. Game theory: mathematical 

models of cor~flict. New York: 

Yalsted Press; 1980. 309 p. 

Swindel, Benee F. 

Theory of games and applications in forestry. Gen. Tech. 

Rep. SE-26, Asheville, NC: [J.S. Department of Agriculture, Forest Service, Southeds tern Forest Experiment 

Station; 1984. 13 p. 

Theory has been developed covering two-person, constant- I 

sum games of strategy. This theory is summarized and 

some possible applications in forestry are suggested. I 

KEYWORDS: Management strategy, zero-sum games, decision 

theory. I 

Swindel, Benee F. 

I 

Theory of games and applications in forestry, Gen. Tech. 

Rep. SE-26. Asheville, NC: U.S. Department of Agriculture, Forest Service, Southeastern Forest Experiment 

Station; 1984. 13 p. I 

Theory has been developed covering two-person, constantsum games of strategy. This theory is summarized and 

some possible applications in forestry are suggested. 

KEYWORDS : Management s trateqy , zero-sum qarnes , decision 

theory. 
The Forest Serv~ce, U.S. Department of Agriculture, i\ cfecficated to the pnncrple of multiple 

u\e management of the Nation'., 

forest resource< for \u\tarned 

yield\ of ~toud. water, forage. 

wildlife. and recreation. Through 

fore,try research, cooperatlon 

uith the States and private forest 

ownerc, and management of the 

Nat~onal Fore5ts and National 

Grasslands, it rtrivect-as directed by Congres+--to provide 

increasingly greater service to a 

growing Nation. 

USDA policy does not permit discrimination because of 

race, color. national origin, sex or religion. Any person 

who believes he or she has been discriminated against in 

any USDA-related activity should write immediately to 

the Secretary of Agriculture. Washington, D.C. 20250. 

s U.S. GOVERNMENT PRINTING OFFICE. 1984-750,