Forest Faunal Systems

Chapter 17

Applications of Game Theory Within Faunal System Management

This chapter shows how a major process can be used to improve decision making, the very foundation of modern faunal resource system management. Game Theory, only remotely related to animals sought by hunters, is a general process used in many fields, including, broadly, economics. It is part of "operations research" and is often cited as one of the major tools of those taking a systems approach.

Game theory is the discipline growing from the mathematical theory of games of strategy first elaborated by Von Neumann and Morgenstern (1944) in Theory of Games and Economic Behavior. The study of decision-making by participants in a competitive environment, it is jointly claimed within university departments of mathematics, operations research, economics, political science, sociology, and others (Williams 1954, Shubik 1955, Luce and Raiffa 1957, Karlin 1959, Thrall 1965, Bartos 1967, Borch 1968, Lucas 1971, Swindel and Yandel 1972, Dukes and Seidner 1978). The degree of common interest is like that exhibited in "expert systems" in the 1980's. Some may view the study of expert systems as the evolutionary sequel to game theory. A journal marking a stage of maturity of the field appeared in 1971 (International Journal of Game Theory). Martin and Sendak (1973) listed 23 sources and applications in forestry. No matter what its parentage or nursery, game theory, itself a study of utility, may have utility for faunal resource management. It attempts to explain observations of systems performance and to formulate principles to guide intelligent action. Shapley and Shubik (1969) described its potential role in analyzing environmental problems. The game model describes in detail the potential payoffs which a person or agency can expect, and it points out how to act in order to arrive at the best possible payoffs in light of the options available to opponents (Lucas 1971). Its utility lies in both analysis and design (Chapter 1, Figure 1.6). It provides the potential basis for developing an alternative theory of faunal resource management since evolution may be seen as"play" of a species against nature, and management as play against animals,"animal space," or user groups. I remember well first meeting "game theory." "This subject describes, in general, the major processes in which I am engaged as a faunal system manager!"

To use game theory, the faunal system manager may conceive of problems, conflicts, or situations as games. No connotation of fun is needed. Two organizational giants may be in a contest to gain a clientele, or their membership funds, or to have a"say" over how some national forest lands are utilized. This contest is a"game." Checkers and chess are games. The situation is one in which there are two or more players, a conflict of interest, multiple choices, rules or policies by which choices can be or are made, and known payoffs. A dog fight is difficult to conceive as a game. There is conflict, yes, but game, no. One act leads to another; there is little or no display of strategy.

Characteristics of a game seem to qualify the following faunally-related situations as games:

1. Anatomical systems play against each other to assure adequate payoffs in energy from a limited energy intake, given certain rules for partitioning proteins.

2. The adreno-pituitary systems of people and beasts respond to a stressful situation, playing either a flight or a fight strategy. The payoff may be death, wounds, or if the stressor is weak, loss of available metabolic energy. Such loss may determine the resources available in the next encounter. When an animal is energy deficient, the same animal that would once fight is likely to take flight. Although no human decision maker is involved, there is a set of decision rules. Analysis of the system as a game will likely provide answers not heretofore possible using a deductive model and statistical sampling.

3. The animal plays against the food supply, spending energy in movement or migration to assure ample or net positive intake. The forage plants play various distribution patterns, growth rates, or anatomical ploys (perhaps spininess or toxicants) to escape consumption.

4. Evolution itself is seen by some theorists as the long-term best coherent play of a species population in a game against multiple opponents to achieve a long-term net energy budget constrained by the requirement to reproduce.

5. The predator plays against the prey.

6. The wildlife habitat manager plants and fertilizes in a play against precipitation.

7. The forester or range manager develops a fire fighting system in a play against"nature" or against all of the causes of fires (Schultz 1966, Thompson 1968).

8. The wildlife law enforcement agent plays against the poacher.

9. The wildlife manager plays against nature, particularly natality and hunting-season weather, as he or she makes recommendations for hunting season regulations.

10. The wildlife manager plays against the agency for salary, promotions, and recognition.

11. The wildlife commissioner, perhaps in coalition with other commissioners, plays against other commissioners in a decision to acquire an area for faunal management or to set a particular hunting season.

12. The wildlife commission plays against the legislature or the governor to achieve a particular act or policy.

13. An author of a wildlife text plays against readers (or a book market) to win. The intended or desired payoff is improved faunal resource management. The potentials are reduced management (e.g., due to inadequate or improper instructions), or a draw. One play, completely "right", may result in no readers; an alternative play or method of presentation may result in less author satisfaction, less quality performance on the land, but many more readers - a few achieving some resource improvements and include human benefits.

The parallels, the rich mathematical and computational developments available, and the logical niceness of the concept of game theory should have great appeal to faunal resource managers. Of note is that a game can be a statement of normative behavior. It thus provides a benchmark for comparing rationality and game-playing effectiveness. Appealing for others, however, is merely the alternative perspective provided by the theory, one that unites them with other problem solvers, worldwide. It is a rich feeling, as if standing on a new peak, to know that faunal resource system management is the complex play of an n-person, non-zero sum game with objectives of choosing optimal courses of action that take into account nature, the possible actions of participants, and chance events. The wildlifer is thus cast as the strategist, the master decision maker, in the roll of ecosystem generalship. The strategist analyzes the game, selects from among the strategies available, and in the dynamically changing situation, makes choices to win most, over the long run.

Objective weighting (Chapter 4 in forming B* and in CAP685) can be conceived as a gaming technique in which a choice must be made between alternatives. There is one active player against all others who are unnamed.The payoffs are estimated, and the decision maker adopts a strategy of choice as maximizing (or minimizing) some index of benefit or effectiveness resulting from his or her choice. This is called a game played under the conditions of certainty. The phrase "conditions of certainty" is misleading for I have never conceived of estimates of effectiveness as"certain." Certainty applies more to the alternatives than to the outcomes. It relates more to the rules of the game and the plays than to the payoffs. The reason "certainty" is employed seems to be that a player is assumed to know, at least tentatively, what a machine, practice, or policy will do. He or she is willing, at least, to "make a stand" that there will be a state of nature, S_i. The player may have only an estimate of the degree or amount of outcome, but at least will not be surprised. The background premise is that given enough time or funds, he or she could be very certain of the values to enter into the effectiveness portion of the objective-weighting matrix. The manager evaluates the various alternative states of nature resulting from each potential decision and selects the desired probabilities to such moves. An amazingly large number of forest faunal system management decisions can be categorized as being made under this formal designation of "certainty."

Many wildlife management games, however, are developed under conditions, said to be, of risk or uncertainty (Halter and Dean 1971). Buffington (1972:37-50) sought to clarify these terms in constructing an information system for aiding decision making in the U.S. Wildlife Refuge System. He said that decisions made under risk are those for which two or more states of nature are possible and the probability of each can be calculated or assigned. Decisions made under uncertainty, in contrast, are those for which the decision maker does not know the likelihood of occurrence of various possible states of nature. The condition will be discussed later.

Buffington used the general decision matrix of Table 17.1 and defined, as follows:

A₁'s = alternatives or courses of action available in a particular decision situation (i.e., tactics or strategies, defined later)

S_j's = states of nature (the configuration of the largely uncontrollable variable)

V = value associated by the decision maker with each outcome expressed as monetary value, a utility scale, or rank order scale (Emory and Niland 1968:274)

x_ij's = evaluated outcomes of matrix A, i.e., payoffs = V(O_ij)

Table 17.1 The decision matrix of standard decision theory. Matrix A is an outcome matrix which is transformed through a value system to matrix B, the payoff or measure-of-effectiveness matrix. (Symbols are explained in the text.)
Matrix A -- States
Alternatives A1 A2 . . Am	S1 O11 O21 . . Om1	S2 O12 O22 . . .	. . . . .	. . . . .	Sn O1n . . . Omn
Oij = f(Ai, Sj)
Matrix B -- States
Alternatives A1 A2 . . Am	S1 X11 X21 . . Xm1	S2 X12 X22 . . .	. . . . .	. . . . .	Sn X1n . . . Xmn
Xij = V(Oij)

Under the condition known as risk, the decision maker does not know the true state of nature but rather has partial information which can be expressed in terms of probabilities applicable to all of the possible states of nature. The probability associated with each possible state of nature may be derived either objectively or subjectively (Morris 1964:248, Richards and Greenlaw 1966:46). I recognize this is a contested proposition. In decision making under risk, the matrix in Table 17.2 is used and includes the probabilities P(S_j) of each state of nature (S_i). The expected value E(A_i) of each alternative A_i is the sum of the probability-weighted payoffs. Thus: E(A_i) = P(S₁)(X_i1) + P(S₂)(X_i2) + . . . + P(S_n)(X_in)

Table 17.2 The decision matrix in a decision under risk problem. Symbols are explained in the text.
States
Alternatives	S1 P(S1)	S2 P(S2)	.	.	Sn P(n)
A1 A2 . . Am	X11 X21 . . .	X12 X22 . . .	. . . . .	. . . . .	X1n . . . Xm	E(A1) E(A2) . . E(Am)
E(A1) = P(S1)X11 + P(S2)X12 + ... + P(Sn)X1n

The wildlife manager selects the optimal alternative, the one with the largest expected value:

E(A_i) = Max E(A_i)

Most forest faunal resource system management decisions seem to me to be made under risk.

A fox knows in what direction the rabbit's next dash may be (stop, left, ahead, or right). He can move left, and if the rabbit dashes left he will intercept. If he moves left and the rabbit right, he will reduce the chances of capture. He may play a"stay on the track" game, unwilling to play for either an easy kill or a sure escape. This fox-rabbit example is too simplistic since many decisions are made in a split second. They are routine, unimportant, pre-formed, random, and rarely involve many comparisons of alternatives and consequences. While game theory is suited for major decision, the frequency with which the unimportant decisions seem to cause human failure makes all efforts toward improved decisions increasingly important.

Decisions made under uncertainty, as stated above, are those for which the decision maker does not know the likelihood of occurrence of various possible states of nature. Each course of action will lead to one of several possible specified outcomes. The decision maker cannot estimate the probability associated with a state of nature. Such a condition for the modern faunal system manager, is almost inconceivable to me, but I presume that it can occur. For the arch-conservative decision maker, a person who does not know what he or she knows, it will replace decision making under risk.

Reaching a solution for a decision under uncertainty requires the decision maker to select an alternative based on payoffs as shown in Table 17.2. Games, by definition, do not (without special provision) accommodate the unique event, such as effects on habitat of a UFO exhaust or the stream pollution caused by a Martian troop bivouac. Games are analyzable when the decision maker knows or can ascertain all of the relevant alternatives and states of nature and can compute the payoff (or an estimate of it) for each combination of state of nature and alternative.

There is implicit in game theory that all players are rational. Readers who have been to public wildlife hearings or to a wildlife research fund allocation meeting may find such an assumption difficult to accept, but the difficulty can be overcome by definition. There appears to be cognitive limits on human rationality but within these limits, largely due to limits in human ability to handle the complexity of many situations, human choice is rational (Laing and Morrison 1974:195). This seems to be the basis for Boulding's (1966) doubt that even if people can meet the theoretical requirements of the rational person, they would probably behave in an irrational manner. The criteria for interpreting such behavior, it seems to me, are usually unavailable. There are certain games that are so large, complex, or with such unique characteristics that there are no known solutions. Why should a player in such a game be judged irrational? Why should a person without infinite memory of all past plays and positive knowledge of future expectations be judged irrational? I find the argument of"rationality" like one of "efficiency." An efficiency of 1.0 exists by definition. Although no engine reaches that level, such failure does not deny the existence of the concept or its usefulness. Halter and Dean (1971) concluded their book on decision making under uncertainty with the disclaimer:

If the decision makers use the decision framework of this book, all of their decisions will be 'good' in the sense of maximizing expected utility. Unfortunately, we are still dealing with uncertainty, and a carefully reasoned decision might still have a bad outcome in any particular instance. We do not guarantee good outcomes - just good decisions!

For those who find rationality unacceptable as a basis of operation, Simon's (1957) concept of the natural person may have appeal. This person is intendedly rational, has only limited knowledge, and makes decisions based on simplified models which lead to satisfying rather than maximizing. Von Bertalanffy (1968:23) said, "Game theory was hopefully applied to war and politics; but one hardly feels that it has led to an improvement of political decisions and the state of the world; a failure not unexpected when considering how little the powers-that-be resemble the 'rational' players of game theory." Generally, rational behavior means that each player, given two choices, will select one that is preferred, that is the one likely to yield the greatest benefits to the person (to maximize utility). Richmond (1968:528) said people will seek to maximize (or minimize, whichever is relevant) the expected value of the objective function. Players choose on the basis of achieving the most likely desirable consequences. When two absolutely equal choices are given, the rational person will toss a coin to make a choice or simply say the situation is moot and let someone else decide. The converse to rationality is irrationality or capriciousness. (Even these, eventually, might be analyzed as a particular "rational" strategy, no matter how perverse they may appear, and a counter-strategy developed.)

My observations have been that indeed the players in faunal resource management do perform rationally (as suggested above). Given incomplete or false information about what payoffs are likely to be [value of an objective x (1 - risk)], and given the findings by Slovic and Tversky (1974) that players will reject a clearly optimum decision principle if they do not understand it, it seems that game theory provides a realistic model of decision making and thus the basis for continued analysis.

Two words will be useful to help the wildlife manager employ the concepts and tools of game theory. These words are tactics and strategy. A tactic is a move to achieve an immediate end, to effect some object, to achieve a part of an answer or a task. Tactics differ from strategies which are concerned with larger, more encompassing, and long-range movements. A strategy is a prescription for a framework of tactics in every possible situation. Strategies are decisions for generals, tactics for lieutenants. A stratagem is a less encompassing term meaning any maneuver or action designed to deceive or outwit an opponent. It is a diversion, a deception, and a particular type of tactic. It may have the scope of a strategy but, if so, it is then usually a piece of a larger strategy.

There are four major "name" strategies for rational game players. These will be outlined later. It will be useful at this time, however, to recognize that there are pure and mixed strategies. A pure strategy is a decision to select always the same course of action. Among four choices the probability of a win may be designated p = (0, 0, 1, 0). This means that the probability of a win with the third choice is 1, the probabilities for 1, 2, and 4 are zero. The reasonable player will always choose the third alternative. This is rational if you are dominant, a constant winner, or if the opponent clearly has such a strategy in a break-even game. Not intuitively apparent is that one pure strategy, tossing an honest coin, may be the best possible strategy to follow. This leaves the manager's choice to chance. If the manager can see one choice that is definitely best, then the opponent will also know this and choose an action to counteract his choice. Thus, using the true coin is a rational approach.

A mixed strategy is a decision to choose at least two courses of action with fixed probabilities. The advantage of doing so is that an opponent is always kept guessing as to which course of action is to be selected on any occasion. Many faunal system managers will recognize that wildlife administrators seem to use this strategy, never having named it.

In playing a mixed strategy, a possible set of four game outcomes or payoffs (p) might be imagined to be p = (2, 5, 1, 0). The player will choose alternative one two-eights of the time, alternative two five-eights of the time, alternative three one-eighth of the time, and he or she will never choose the fourth alternative (CAP9300).

There are numerous kinds of games of varying difficulties which can be applied to many problems. Beyond the scope of this chapter, games are variously classified by (1) number of players, (2) number of moves and choices, (3) constant-sum and general-sum cases, (4) cooperative situations, (5) states of information available to players, (6) and restrictions placed on side payments. These are overlain by a taxonomy of extensive, characteristics, or normal forms (Lucas 1971:2). One of the simplest and most commonly referred to games, is the two-person zero-sum game. Such a game played under certainty involves two opposing players, the objective of each being to create as large a gain as possible on each play of the game. Later the"gain" may be translated into a simple win or loss. What is gained by one player (e.g., +3) is lost by the other (e.g., -3), (one player gains $3, the other loses $3; ((3) + (-3)) = 0, thus it is called a zero-sum game, in which each player competes for the largest share of the"kitty" (as in a forest faunal research budget announced within an agency), and also the n-person zero-sum game (Rappoport 1970, Rappoport and Chammah 1965), within which there is the possibility of the players forming coalitions. ("If we team up, we will win over A. We'll have to divide the winning but an almost sure 'half' is better than a 50-50 chance of getting nothing.")

With the terms and major concepts outlined, it is now possible to describe the four major game playing strategies. These are

• maximax (or minimin),
• minimax (or maximin),
• Hurwicz, and
• Savage or minimax-regret.

The maximax strategy has been discussed previously under objective weighting (Chapter 4) and above under games having conditions of certainty. A choice is made from among alternatives based on some index of maximum utility or expected value. The maximax is played, expecting the best possible state of nature and therefore the decision maker selects the alternative that provides the maximum payoff. (It can be played, as the other side of the coin, the minimize loss - thus a minimin strategy.) The maximax is usually viewed as adventurous or a Polly Anna strategy.

Strategies for playing a simple game can usually be determined by visually inspecting a payoff matrix associated with the possible choices. For a game involving three alternatives for each player, a possible payoff matrix is shown in Table 17.3. (See CAP5010.) In this matrix, three units of payoff (which could be in units of thousands of dollars, cohorts of trophy animals, or megacalories of forage) is both the minimum value of row three and the maximum value of column one. This value is called the saddle point (like a topographic saddle, the low place between two hills, a pass.) The player who has the rows (a) will always choose alternative row three (since he can gain at least three units). He or she chooses the highest of the lows. Thus, from potential strategies p = (a₁, a₂, a₃), he or she has an optimum strategy p# = (0, 0, 1). That player with the columns (B) will always choose column one since he or she is guaranteed not to lose more than three. The player's optimum strategy is q* = (1, 0, 0). Since one player will gain three and the other lose three, if both choose their optimum strategy, the value of the game (V#) is said to be three.

Table 17.3 An analysis of a two-person game can be made as a payoff matrix. The payoffs to Player B are shown, given Player A and Player B will chose one of the alternatives.
		Player B
		b1	b2	b3
Player A	a1	0	-8	6	-8	Row Minima
	a2	2	7	-1	-1
	a3	3	9	4	3*
Column Maxima		3**	9	6
*The maximum of the row minima is 3. **The minimum of the column maxima is 3.

If the players in the preceding example both chose 3, they would have used the minimax or the maximin strategy. Player A determines the minimum gains associated with each alternative and then chooses the action which produces the maximum of these minimum gains (the maximin strategy). Player B then determines the maximum loss for each action and chooses the minimum of these maximum losses for his play of the game (the minimax strategy).

Which strategy should be used? There is no singular answer. It depends... The minimax strategist seeks the lowest of the large losses. The maximin strategist, a conservative fellow, seeks the highest minimum gains. When a saddle point exists (you can tell by using the rhyme: when the high in the column is like the low in the row), then neither player has anything to gain by departing from the strategy which leads to the saddle point. When a committee seems"stuck," or arbitration has ceased, or when a discussion"just can't get off dead center," it is likely a saddle point situation exists. A minimax strategy may not be very satisfying to the bold, but it is safe. It is a decision to seek a payoff that will never be worsened by the opponent's choice. Neither player can improve his or her position by selecting any other strategy. In the above example, player A thinks: What's the smallest I can lose with each play? Strategy a₁ (0, -8, 6) guarantees a minimum loss of -8 (because a gain to B is a loss to A); to a₂, -1; and to a₃, a minimum outcome of 3. Of these minimum outcomes, he or she selects the maximum (marked with *), a maximin choice, and the minimax (or upper) value of the game is 3. In other words, the player adopts the minimax criterion of optimality, observes the feasible strategies, and then chooses the alternative which results in the best of the worst outcomes. This player is a shrewd fellow who, no matter what plays are made by his or her opponent, will keep the gains from such plays to a minimum.

In the above game, it is easy to see that if a player does not know the payoff matrix, it is difficult for him or her to make any decision except by guessing. Absolute knowledge is not necessary because it will often be seen from using estimates in the analysis that there is a wide range over which one strategy will win. Also, if the player has a priori robabilities, he or she will make the optimum choice all of the time and part of the decision making activity is lost. If the player knew his or her opponent's choice, he or she would not be"fair," since he or she would always score large gains. One role of research is to learn as much as possible about Nature's plays, thereby making the game as "unfair" as possible. Research allows winning plays against nature. Games do not have to be "fair."

There is an assumption in the games discussed in this chapter that the utility function and the parts of the payoff matrix are approximately linearly related, since it is improper to add or multiply units that change at different rates depending on the play. Advanced developments in theory deal with this very harsh constraint but it is unimportant in annual faunal system games.

If the payoff matrix for a game does not have a saddle point, then finding the value of the game is more difficult and knowledge of some equations of the theory is necessary. Finding the value of the game is only part of the solution, however. Actually to solve a game means to find not only the value of the game, but also all pairs of optimal strategies. The solution to a game is then given by the set (p, q, V). To determine these values for a matrix without a saddle point requires using the following formulae which are adapted to a two-by-two matrix (Bartos 1967:168).

v  =

a b c d

p₁ = (d-c) / (a + d - b - c)

p₁ = (probability for A to choose strategy 1, probability (1-p) to choose 2)

p₂ = (d-b) / (a + d - b - c)

v = (ad - bc) / (a + d - b - c)

These equations make it possible to solve all games that frequently occur involving two alternatives or alternative groups after lesser alternatives have been discarded. Each player has two alternatives and there is a definite payoff matrix. (Where multiple alternatives seem to be involved, they may often be narrowed to some final critical pair of choices.) The results of the analysis are weighted averages of the expected payoffs.

An example may be helpful. There is a faunal resource manager who has a large area with a dense deer population. There is no access to the center of the area and hunting pressure is high around the outer edges. The area is an excellent deer habitat and the population increasing rapidly. The manager, after a study, realizes a decision must be made between two choices:

1. Build roads and create access to the center of the area. This would increase hunter success and would also probably increase the number of hunters (and person hours). By allowing harvest, the manager may also be able to stabilize the population of deer at the available forage supply.
2. Significantly improve deer habitat around the edges of the area in order to increase the number of deer in the outer areas, thereby increasing hunter success and satisfaction.

Which play should the manager select? By using game theory and assigning values to all the possible outcomes so that the values reflect the preference of outcomes, the problem may be analyzed within the following matrix and Table 17.4. Payoffs are in mean annual thousands of person hours of hunting recreation (over 5 years). Since a saddle point does not exist, applying the above equations for a mixed strategy yields answers as follows:

Table 17.4. A game matrix of a play between "the manager" and "the area" with payoffs for the situation described in the text.
		The Area Responses
		1 - Population Stability	2 - Population Increase
The Manager's Choices	1 - Roads 2 - Habitats	13 14	18 12

p = (12 - 18)/ (13 + 12 - 18 - 14) , (13 - 18) / (13 + 12 - 18 -14)

p = +0.29, + 0.71

q = (12 - 18) / (13 + 12 - 18 - 14) , (13 - 14) / (13 + 12 - 18 - 14)

q = +0.86, + 0.14

v = ((13 x 12) - (18 x 14)) / (13 + 12 - 18 - 14)

v = 13.7

This means that the manager should play the road-building strategy 0.29 of the time and the habitat improvement strategy 0.71 of the time. The latter makes sense and implies that, for example, by using random numbers (CAP112), he should go with "habitat improvement" when a number between 0-70 comes up, with"roads" when 71-99 comes up. This will guarantee a payoff of at least 13.7 thousand person-hours of recreation a year.

There is an alternative way of using this same concept, not to find the perfect strategy, but to get an idea of the limits to some probabilities and the frequencies with which actions might be taken. The faunal system manager typically asks: will I win most of the time if some value (that I can determine) is not exceeded? The value might represent a rainfall as the manager builds a forest pond, a desired soil moisture relative to a plant response to fertilizer, or a weight assigned to a game season objective.

Take the situation of two rainfalls as nature's plays and fertilization of a game bird food plot as the manager's play. It might appear as in Table 17.5. Payoffs are in grams per plot relative to some base datum of goodness or sufficiency. This game has a saddle point (3 is high in the column, low in the row) and there is no real question about an optimum strategy for the manager; it is to employ the low fertilizer requirement. Perhaps the manager is unsure of the rainfall and wants to know at what level he or she should shift from a low to a high fertilizer strategy.

Table 17.5. A game matrix for manager against nature. The manager's decision options are the amounts of fertilizer to be applied to achieve game bird food yields.
		Nature
		High Rain	Low Rain
Manager	High Fertilization Low Fertilization	10 6	-1 3

(Ackoff 1962:59). Suppose nature were to select "high rain" with a probability p and "low rain" with a probability 1-p. Then if the manager selects high fertilizer, his or her expected payoff is (p)(10) + (1-p)(-1), and if he or she chooses low fertilizer, the expected payoff is (p)(6) + (1-p)(3). The manager knows that the low fertilizer choice is best so long as:

(p)(6) + (1-p)(3) > (p)(10) + (1-p)(-1)

6p + 3 - 3p > 10p - 1 + p

3p + 3 > 11 p - 1

p > 4/8 or 0.50

As long as the probability is greater than 0.50 that nature selects the low rainfall, the manager should select the low fertilizer. The 0.5 is a restatement of the saddle point concept. Now suppose a saddle point did not exist, for example, because of a peculiar water and nitrogen interaction. The table of relations under such a condition might then be:

v =

10	-1
2	3

Following the same line of reasoning but with a high fertilizer strategy likely to have the highest yields, then:

(p)(10) + (1-p)(-1) > (p)(2) + (1-p)(3)

p > 4/12 or 0.33

In this new situation, the manager should select strategy a₁, the high fertilizer, as long as nature plays the high rain strategy, (b₁), more than 0.33 of the time. If less than 0.33, the manager should select the low fertilizer strategy. For example, if high rainfall occurred only 0.29 of the time (e.g., due to topography or regional norms), then the expected value of the manager's high fertilizer option is:

(0.29)(10) + (0.71)(-1) = 2.19;

and for the low fertilizer option:

(0.29)(2) + (0.71)(3) = 2.71.

Clearly the expected returns from a less costly option are greater and the manager should apply the small amount of fertilizer. How does a manager determine such probabilities? The published research data can be useful. Research may (and should) be directed to obtain answers to fill such tables. The alternative is to estimate, e.g., to ask and answer how many times out of 1000 will x occur? The results from experts are likely to be helpful and instructive.

The two-person non-zero-sum game under risk can be demonstrated by the two-wildlifer game in which both play to attend a national wildlife scientific conference at agency expense.

Because of primitive agency policy, only one of the two can go in any year, and only if they are presenting a paper. Therefore chances of attending for each of the two wildlifers are 0.5, 0.5. The chances for acceptance of a paper are 0.50 based on past records. (A very large number of papers are submitted.) Thus the chances for probable payoff, no matter what the size, to two successful wildlifers is 0.25, 0.25. The payoff analysis of the game is shown in Table 17.6.

Table 17.6. The analysis of the possible gains in a two-person non-zero-sum game. Frida's and Butch's net likely gains are expressed in terms of an index of salary, promotion, receiving choice assignments, and tendency of their supervisor to overlook minor mistakes and inefficiencies.
		Butch
		b1 Submit a Paper	b2 Do not submit a paper
Frida	a1Submit a paper a2Do not submit a paper	6,6 1,7	7,1 2,2

The game, in which cooperation is not a feasible option, is analyzed as follows for Frida. The same analysis holds for Butch. If Frida submits a paper, she looks good in the eyes of her superiors. ["You've got to give her credit for trying," an absurd slogan.] In addition, Butch will have to perform stand-in work for her while she is away. However, he did not waste any time writing a paper or experience the associated pain, so he has a net gain. The first simple payoff matrix (Table 17.6) must be modifed on the basis of the odds of the paper being accepted (if submitted); i.e.,

6,6	7,1		0.5	0.5		3,3	3.5,1
		x			=
1,7	2,2		0	0.5		1,3.5	2,2

Next, it is modified to to agency policy, i.e., that only one may attend even if both papers are accepted. This policy results in the matrix:

3,3	3.5,1		0.5,0.5	0,0		1.5,1.5	3.5,1
		x			=
1,3.5	2,2		0,0	0,0		1.3.5	2,2

Frida's likely gains in the play against"The Conference" are shown in Table 17.7.

Table 17.7. A game matrix for the manager, Frida, against the conference.
		Conference
		a1	a2
Frida	a1 a2	1.5 1	3.5 2	1.5 1	Row Minima
Column Maxima		1.5	3.5

It is rational for Frida to submit a paper (but not by much). She would employ a maximin strategy, selecting the maximum expected payoff (1.5) from the row minima. It is rational, but it is clearly in the best interest of the players to change the rules of the game! At least they could play a correlated mixed strategy and, flip a coin to see who works on and submits a paper each year. Thus, they will win much more over the long run.

The maximin or minimax has been emphasized as a decision making and game-playing strategy. The Hurwicz strategy (or the optimist's strategy) is a viable option. It allows an additional weight to be brought into the decision.

Suppose you were asked to advise Butch and Frida. Select a value of W (where 0 < W < 1) which is a measure of the player's optimism. Then operating in the rows, select the smallest component (e.g., in row 1 it is 1.5) and call it a. Select the largest in the row (e.g., 3.5), and call it A. (There can be n rows and columns.) Then compute for each ith row, a_1* = A + (1 - W)a. The best alternative is the one with the maximum value for a_1*. In the above case where W is estimated to be 0.60 for Frida who is "gung-ho", the results are a_1* = 2.7 and a_2* = 1.6 and the advice is for Frida to get to work on a paper. It can be seen from the above analyses why, given certain values, the rational manager would not submit a paper.

The Hurwicz strategy, as I see it, falls between the maximin strategy and the maximax strategy. The latter is as adventurous as the maximin is conservative. It is too pessimistic to assume that the worst will always happen.

An alternative strategy for game playing is suggested in the agent-poacher game. This is a two-person, zero-sum game. A hypothetical game is shown in Table 17.8.

Table 17.8. Description of the payoffs in an hypothetical deer poacher and law enforcement agent's game. The minima and maxima are evaluated for the agent's decision.
		Poacher or Spotlighter
	1 Moving car patrol - Area 1	4,2	-2,6	-2 Minima
	2 Moving car patrol - Area 2	0,8	5,2	0
Agent	3 Stake out - Area 1	6,1	-2,6	-2
	4 Stake out - Area 2	0,8	8,1	0
	5 Stay at home	0,8	-2,6	-2
		Max. 6	8

The wildlife law enforcement agent has five strategies, the poacher two. The agent's gains are positive resource yields. The losses are those to sensitive deer herds. For a poacher to go to road 2 while the agent is elsewhere results in a loss of 2 (-2) because the herd nearby is in better shape than in the vicinity of road 1. The herd near road 1 has surplus animals, the range is heavily browsed, and the poacher has no significant effect. Moving car patrols are less effective than stake outs. To stake out an area is to sit in a car with an assistant and observe a large area, then rush out to apprehend a poacher who shines a spotlight from a car over a field to "catch" the light-reflecting eyes of deer or elk and then make a rifle kill. The maximum value of this game is one which will maximize the minimum gains to the agent. A saddle point does not exist, thus the strategy of staking out area No. 1 about 0.4 of the time 6/(6+8) and area No. 2 about 0.6 of the time is rational. The above strategy guarantees the best of the worst possible results.

Without any knowledge of the poacher, the Laplace strategy assumes all outcomes are equally possible and therefore the wildlifer should select the row with the maximum average payoffs, i.e., a_**i = (a_i1 + a_i2 + . . . +a_in) n and the solution is Max a_**i.

A last strategy has great appeal. It is called the Savage or Minimax-Regret strategy. The idea is to minimize the decision maker's maximum regret if the choice he or she makes is wrong. It is done by redefining a matrix as a regret matrix. You select in each row the maximum gain, then subtract the maximum in that row from all of the related gains in other rows. For example, the matrix:

0	50
10	5

transformed to a regret matrix is

10	0
0	45

It shows the difference between the payoffs obtained and what would have been obtained if the largest payoff had been selected (Ackoff 1962:51-52, Thrall 1965:19-23). The maximin strategy is then applied by player A to this matrix:

A	B
	10	0	10 Row Maxima
	0	45	45

Player A selects the alternative for which the maximum regret is minimum, i.e., a₁ or 10.

What is the best of the above four strategies to use? It depends. There is yet no known ultimate criterion for selecting among the four strategies described. As in statistics, when confidence level must be decided, the decision maker is in charge and must establish confidence limits by means quite apart from the numbers being analyzed. The choice of a strategy is for the strategist. How sure the faunal system manager must be, how prone he or she is to risk taking, how short or long their concept of time, how passionate they are to win, how frequently stratagems are used, and how resilient they think a person, organization, or an ecosystem will be - all determine the appropriate strategy.

In the future, game theory may be applied to natural resource problems. Faunal resource system management can benefit. There is little proof that game theory is useful. Lucas (1971) observed that in his 2.5 years with RAND Corporation among a dozen game theorists, he was consulted several times each month about an important, non-obvious, real-world problem for which the game theoretic approach was the most appropriate analytical tool. There are major stumbling blocks, and even more practical hurdles like users, and available data. In some situations when all possible strategies for two players (and worse for n-persons) are laid out, solutions using a game theoretic approach may have to be abandoned because the problem, while computationally feasible, is too data hungry. I suspect there will be found ways around such problems - ways of using estimates, examining strategic sensitivity, the relevant scope of play. Expert system development with executive estimates provides an alternative structure for using many of the concepts outlined here.

There are useful extensions already available in continuous games (Subert 1967) (as would be the case in one wildlifer being employed after another as responsible for a management area, or as society trying to win against nature on a playing board wrecked by a previous society). Walsh (1970) and Walsh and Kelleher (1970) presented the concept of a median value to replace the expected value as the basis for play in two-person games. The use of the Beta or Weibul distribution for generating such values seems likely to be useful. The nucleolus is a term (Schmeildler 1969) of n-person game theory that means the play which successively minimizes the maximum excess (the loudest complaint). That such a term has intuitive relevance to wildlife management problems suggests a need for investigating it and attempting to develop a practical, functional, computer-based methodology.

I have observed that most wildlifers perform as if they were in "check," as in chess. Buffington (1972) discussed a constrained decision algorithm. In essence, there is only one residual alternative after all constraints have been applied and thus there is no decision. (A decision seems to be being made as the constraints are listed and realized. A choice can only be made between two or more alternatives.) Under complete constraint, there is not even the go, no-go option. "In the end," said Buffington (1972) of such situations " the decision maker's role is to announce the decision already made by the constraining action." The wildlifer can be subject to such constraints (in my view, the usual case) or can create them. Policy, precedence, law, directives, public involvement, peer pressure, and press coverage are all means for constraining decisions. Many games cannot be won; they must be played...to prevent defeat. Other desired outcomes for playing typical realistic wildlife games are minimizing the duration of resource conflict, or minimizing the difference between large numbers of antagonists and protagonists (the concept of stable complaint). One of the truly great needs is the end-game gambit. Many managers of the forest faunal resource, like the rest of society, either do not know when to quit or how to get out of a game. They, as victims of the old joke, observe,"Yes, I know the game is crooked, but it's the only one in town!" Playing so that you may play for higher stakes later may also be a strategy, but losses build rapidly and the "big game" may never come.

As suggested in this unit, game theory can be applied to many levels in faunal system management. After presenting an analysis of forest nursery practice as a game against nature Jeffers (1966) (see also Jeffers 1978), presented means for solving the problems of this complex operation. He said,

It is not difficult to see why the model has not so far been applied to the practical problem. It is not that the mathematical mode is difficult to build and implement. It is, rather the nature of the model, and of the solution that it suggests, is so radically different from the methods that are being used at present to provide an answer that it is difficult to see where to begin the process of introducing the model as a practical solution.

Makower and Williamson (1967:209) observed:

This is really the crux of the problem of introducing modern mathematical techniques for the solution of practical problems of industrial and commercial management. In general, the manager of today is non-mathematical, and the resistance to change, which plays a part in the make-up of all of us, may be strengthened by the unfamiliarity of the mathematical concepts which underlie the proposed solutions to his problems.

In everyday life we are constantly making decisions; some trivial ... some not so trivial. Although we rarely, if ever, formulate the decision in detail, as above, we may nevertheless unconsciously carry out a similar exercise. There are decisions, however, where the issues are of such importance as to demand time for quantitative analysis. It is in such situations that the value of the concept of game theory lie. Its ultimate usefulness in practice depends on how exhaustively the courses of action may be determined and how accurately the possible outcomes may be measured. Game theory does at least provide a framework within which the relevant factors involved in a decision may be isolated.

After the game theoretic analysis is complete, it is tempting to conclude, "I knew that anyway." Like the researcher who thinks he or she has observed a significant difference but really cannot tell until performing a t-test, the faunal manager often thinks he or she knows the optimum strategy but cannot really tell until after analysis. Although some have claimed game theory offers no new insights (Lucas 1971:6), it appears that judgement may be premature. It seems to provide a structure for rethinking a complex field like faunal resource system management. If the resulting understanding is more clear, complete, or rich, then game theory will have been useful.

References

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Last revision May 21, 2001.