ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR AN APPLICATION OF GAME THEORY TO SIGNAL DETECTABILITY Technical Report No. 20 Electronic Defense Group Department of Electrical Engineering By: T. GBirdsall Approved by;.. - A. B. Macnee Approved by:' W. W. Peterson Project 1970 TASK ORDER NO. EDG-3 CONTRACT NO. DA-.36-0039 sc..15358 SI( AL CORPS, DEP-kAR'rW fl%4ENT" OF HE ARfltr

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TABLE OF CONTENTS Page ABSTRACT iii ACKNOWLEDGMEENTS iv 1. INTRODUCTION 1 1.1 Remarks 1 1.2 Purpose of a Game-Type Solution 1 1.3 The Problem 4 2. TE: GAME SOLUTIONS 5 2.1 The Game Defined 5 2.2 A Numerical Illustration 7 2.2.1 The Payoff 7 2.2.2 Direct Solution 10 2.5 The General Receiver Solution 13 2.3.1 Derivation and Proof of Solution 13 2.3.2 Existance 17 2.3.3 Summary 18 2,5.4 Use of the R. 09 C. Curves to Obtain Particular Solutions 19 2.4 The General Transmitter Solutions 21 2.4.1 A Point Solution 21 2.42 Uniqueness of the Receiver Solution 22 2.4.3 Distribution Solutions, Non-Uniqueness of Transmitter Solutions 24 2.5 The General Solutions Applied to the Numerical Illustration 25 2.6 Limiting Cases as Detection Improves 28 3. SOLUTIONS WHEN INFORMATION IS AVAILABIE 30 3.1 A Priori Probability Restricted to a Known Interval 30 3,2 A Priori Probability a Random Variable with Known Mean 32 4. SUN4ARY OF SOLUTIONS 33 LIST OF SYMBOLS 35 DISTRIBUTION LIST 37

ABSTRACT The problem of unknown a priori probability P(SN) is treated by considering a zero-sum two-person game, where the payoff to the receiver player depends on correctly detecting signals in noise, his opponent choosing the average Ton the air"' time, and solutions for both players are determined. Completely unknown, known within a range and randomly distributed a priori probability cases are solved. Particular emphasis is placed on the solutions for the receiver. In all instances these specify that the receiver should be of the same type as that specified by EDG Technical Report No. 13 The Theory of Signal Detectability which assumes the a priori probability of a signal's presence is known. The correct operating points of that receiver are specified. iii

AC KNOWLEDGEMES The author wishes to acknowledge his indebtedness to Mr, W. Wt. Peterson for his aid at the conception of this report and to Dr. A. B. Macnee, Dr. H. W. Welch, Jr., Mr W.- W. Peterson, Mr. W. C. Fox for their sugestions the final form of the report. iv

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPLICATION OF GAME THEORY TO SIGNAL DETECTABILITY 1. INTRODUCTION 1.1 Remarks This report is supplementary to Electronic Defense Group Technical Report No. 13, The Theory of Signal Detectability, Part I, in which the general theory of signal detectability is treated. In establishing a definition of optimum in that report, it was assumed whenever necessary that the a priori probability that a signal be transmitted, P(SN), is known. If this assumption does not hold, certain applications cannot be made directly. Several alternative assumptions can be introduced to replace exact knowledge of P(SN). Each of these is based on the idea of weighting the errors and correct responses of the receiver and maximizing in some manner the average or total expected return from these responses. 1.2 Purpose of a Game-Type Solution The amount of basic theory drawn from the theory of games for this report is very small. Primarily, only the definition of a solution of a twoperson zero-sum game is used. It is this definition and its practical value that will be discussed briefly here.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN A two person game consists of the set of rules as to how each player may move, and the rule for paying off at the end of each play of the game. We shall consider games in which each player picks an overall strategy, and the reward to each of the players is given by a payoff function defined for each possible pair of strategies. The game is zero sum when the payoff to one player is the negative of the payoff to the other. Consider the situation in which a particular player's objective can be reduced to the desire to maximize a real function of two variables by controlling the value of only one of these variables. This real function can be considered as his "payoff." The value of the second variable is unknown to him and may be controlled by someone else. It is convenient to assume the latter, and therefore the person in question is confronted with a hypothetical or real opponent, who controls one of the variables of the payoff. For example, let us assume that player A wishes to maximize the value of the payoff V(a,p), and his opponent P wishes to minimize it. For each possible value of p, A can pick his variable "a" so that V is maximized. In general, this value of a is a function of p, that is, a = a(p). After this maximization the payoff is solely a function of p, denoted by either V(a(p),p) or max V(a,p). Player P can then decide which p to pick such that V(a(p),p) is a minimum. Player P does this, and fixes his variable at the value p* accordingly. One can evaluate the payoff V(a(p*),p*); if player A chooses the variable a = a(p*) this will be the payoff, and if player A uses any other value of a, the value of the payoff will not exceed V(a(p*),p*) since by definition a(p*) yields a maximum. This can be condensed into the equation V(a,p*) _ V(a(p*), p*) = min max V(a,p) (1.1) p a 2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If the above procedure is reversed with player P picking the p(a) such that p(a) minimizes V(a,p) for each fixed a, and then player A maximizes over these values choosing a value a*, the result will be max min V(a,p) = V(a*, p(a*)) < V(a*,p) (1.2) a p For the problem considered in this paper; a* = a(p*) and p* = p(a*), the result is V(a,p*) < V(a*,p*) < V(a*,p) (1.3) This is interpreted as follows: calling (a*,p*) the solution, if A plays at the solution and P does not, the value of the payoff is greater than at the solution, and therefore P should have played at p* in order to minimize the value V; conversely, if P plays at the solution and A does not, the payoff will be smaller, and thus A should have played at a*. The solution is an equilibrium point; that is, if both players play at the solution, neither will choose to change. It is called an equilibrium point in order to contrast it with the jockeying for advantage that often occurs: first player A chooses a playing point, then player P chooses his playing point with this in mind, then A changes his playing point to take advantage of P's choice, ad infinitum. Viewed by player A, playing at the solution guarantees him at least a minimum payoff, and he may have even a larger payoff if player P does not choose the solution value. It may well be the case that not only is P at some nonsolution value, but that if A knew this and acted accordingly he could realize a tremendously larger payoff. However, gambling on this latter without real knowledge of P's choice may have the effect of pulling the floor out from under A; that is, he no longer can count on a guaranteed minimum and is liable to much lower payoff value.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The application in Section 2 is to let a player P, either real or hypothetical, choose the a priori probability of there being a signal present. The natural bounds on p are 0 < p < 1 but any smaller bounds can also be chosen (see Section 3). 1.3 The Problem Assume an operator in the field has available a very versatile search receiver, and he is faced with the task of maximizing the return from his operation of that receiver to detect signals of known character in the presence of noise. It will be shown that this operator will require the same type of equipment as an operator who knows the a priori probability in the same situation. Values are attached to each possible action of the receiver operator, and he wishes to obtain the largest average (or total) value. If the following symbols are used to describe events: SN - There is signal and noise present at the receiver input N - There is only noise present at the receiver input (1.4) A - The receiver operator says there is a signal present CA - The receiver operator says there is only noise present then SN*A will represent the event of correct detection, and SN.CA, a miss; N-A, a false alarm; and N.CA, a correct guess that no signal was present. These are the four events that can occur, and to each is attached a value to the receiver player as follows: VSN.A = Value of correctly detecting a signal's presence VN.CA = Value of correctly detecting that no signal was present VN - CA~~~~~~~~~~~~~~'

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN -KSN.CA = Value of the error of missing a signal 1 = Value of the error of falsely saying a signal was present The average or expected value obtained by the receiver operator will then be the sum of these values, weighted by the probability of the respective events. This average is called simply the payoff value V, and V VSNA P(SN.A) + VN.CA P(N-CA) - KSNCA P(SN-CA) - KNA P(N.A) (1.6) 2. THE GAME SOLUTION 2.1 The Game Defined The receiver operator will consider himself as a player in a zero-sum, two-person game. His opponent is equipped with a band-limited transmitter, and he uses the receiver as a detection device. This is enough to allow him to use the general theory presented in Technical Report EDG-13, The Theory of Signal Detectability. Each move of the game consists of a transmission or no transmission for time T by the transmitter of one of a set of possible signals, and the decision by the receiver player in sufficiently small delay time after this so that the values specified are independent of the delay. The game consists of a large number of these moves. For example, T might be a second if listening for Morse code, or a millisecond if looking for radar pulses, and one play of the game would last for a watch, or for the duration of an attack by guided equipment. 1 The peculiar form -K as the value of an error is to emphasize that it is really a loss. For example, if the values are in dollars and cents, and if the cost to the receiver operator is $100 for each miss, then KSN.CA = $100 or -KSN.CA = -$100

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The strategies are as dissimilar as the equipment. The player with the transmitter may choose his average transmitting time, which in the receiver's language is P(SN), the a priori probability that a signal is transmitted. The player with the receiver chooses the subset A of the receiver inputs that he will call signal plus noise, that is, he chooses the "criterion" A. The entire play is submerged in noise that is known equally well to both players by its p probability density function fN(X).l Because the transmitter player chooses his a priori probability only, the signal-plus-noise density function fSN(X)l is made known to both players. In order that the game be realistic, the trivial assumption is made that the receiver operator gains less for either error than he gains from a correct answer. This can be expressed as -K < V for either K and either V, or O<K + V. The expected payoff to the receiver player is the average value V, and to his opponent is -V. V = VSN.A P(SN.A) + VN.CA P(N'CA) -KSN.CA P(SN.CA) - KN.A P(N.A) (1.6) Eq (1.6) can be greatly simplified by the use of conditional probabilities and the fact that SN and N, A and CA are complimentary; that is P(N) = 1 - P(SN) and P(A) = 1 - P(CA) (2.1) Simply substituting so that the terms are in the same order as in (1.6) we have V = VSN.A P(SN) PSN(A) + VN.CA [1 - P(SN)] [1 - PN(A)] -KsN.CA [1- PSN(A)] iKNhA [1 - P(SN)] PN(A) (2.2) 1 The existence of the density function is a basic assumption.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It is convenient to split V into two payoff functions: Vo(A) = V when P(SN) = 0 V1(A) = V when P(SN) = 1 v(A) VNCA [lPN(A)] KN.A PN(A) (2.3) V1(A) = VSN.A PSA) A) -KSNCA [ 1 -PSN(A)] (2.4) Under the restriction that each value -K is less than each value V, neither of these payoffs can dominate the other for all A. This is evident if we consider first A = ~, the empty set1: Vl(0) = KSN-CA Vo(0) = VN.A thus Vo(M) > Vl(o) and second, consider A as all receiver inputs, A = R: V1(R) = VSN.A Vo(R) = -KN.A thus Vo(R) < V1(R). The payoff can then be written V = P(SN) V1(A) + [ 1 - P(SN)] Vo(A) (2.5) 2.2 A Numerical Illustration 2._2.1 The Payoff. The ideas of payoff functions and min-max solutions may be made much clearer by treating a particular numerical example before considering general solutions. Therefore for the sake of example let us pick values for the payoff, To choose the empty set 0 as a criterion means that the receiver operator will never say there is a signal present, i.e., will always say there is noise alone present. The reverse extreme is to choose the whole space R as a criterion.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN vs 7 -K e = -8 VSNA 7 SN CA -K -.6 NCA 2 From (2.3) and (2.4) VO and V1 are computed Vo(A) = 2 [1 - PN(A)] - 6PN(A) = 2 - 8PN(A) (2.3n) Vl(A) 7PSN(A)- 8 [ - PsN(A)] =15 PSN(A) - 8 (2.4n) The first forms above emphasize 2 > Vo(A) > -6 7 2 V(A) - 8 We know from the general theory that for each value of PN(A), all possible values of the probability of detection, PSN(A), lie in an interval, the bounds depending on the types of signals and the noise. Therefore, further assume, again for the sake of example only, that the signals and noise are such the points in the crosshatched area of Fig. 1 represents all of the possible combinations of the probability of detection PSN(A) and probability of false alarm PN(A).

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1.0 0.8 PSN(A)' ~ —~f~ttU ALL POSSIBLE 0.2 0 0.2 0.4 0.6 0.8 1.0 PN (A) FIG I REGION OF ALL POSSIBLE CRITERIA IN A TYPICAL DETECTION CASE For the moment consider only those criteria A with a given false alarm probability, say 40 per cent. From Fig. 1 the limits on probability of detection are seen to range roughly from 11 per cent to 76 per cent. Vo(A) depends only on the probability of false alarm and is this case Vo(A) = -1.20 for all the criteria with 40 per cent false alarm. However, V1(A) ranges from -6.35 to 3.40. Two criteria of interest are those that take on these values: PsN(A) =.11, PN(A) =.40 and S,(A) =.76, PN(A) =.40 By restricting our attention to these special criteria./, and AS, we can plot the payoff V as a function of the probability of false alarm and the a priori probability P(SN). Eq (2.5)shows that V is a linearly combination of Vo(A) and t~~~~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN V1(A). Plotting Vo(A) against PN(A) yields a straight line from VN.CA at PN(A) = 0 to -KN.A at PN(A) = 1. Plotting V1(Aa) and V1(A,) against PN(A) yields 2 curves, V1(kI) below V1(A.), both from -KSN.CA when PN(A) = 0 to VSN.A when PN(A) = 1. If these two plots are made on parallel planes and corresponding parts (same PN(A)) connected by straight lines, the resulting surface will enclose a volume representing all possible values of the payoff V. Because the receiver operator can always operate on the upper surface of this payoff volume by using A. type criteria, he will certainly do so in order to maximize the payoff. This upper surface will be referred to as the receiver's upper payoff surface, and for the numerical case in question is shown in Fig. 2. Figure 2a is a general view of the upper payoff surface. Four curves of special interest are indicated on the orthogonal projections. In Figures 2c and 2d it is apparent that there is a horizontal line on the the surface parallel to the P(SN) axis; this is marked (1) in Figures 2b, 2c, and 2d. The curve marked (2) has constant P(SN) and is a maximum when it crosses curve (1). Later, Fig. 3 on page 12 will show the maximum value of V for each fixed value of P(SN); the maximum values occur along the dotted curve marked (3) in Figure 2b. Figure 3 is actually this curve as it would appear in projection 2c. The final curve consists of the horizontal line (1) and the 2 pieces of curves marked (4); along this curve is the minimum value of V for each fixed value of PN(A), the analog of curve (3). 2.2.2 Solution of the Numerical Case. We have argued in 2.2.1 that the receiver operator would use criteria on the upper curve of Fig. 1. The 10

o - -- 4).1 -.2 -.3. P(SN) O3U.6 0.7-.8 - FIG. 2A FIG. 2B 1.0 1(4)1 — _ I I I I I I I I I 0.1.2.3.4 P(SN).6.7.8.9 1.0 0.1.2.3.4 PN(A).6.7.8.9 1.0 -2V 4 FIG. 2C FIG. 2D FIG. 2. AN UPPER PAYOFF SURFACE.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN question of which point on this upper curve to use would be answered by equation (1.6) of The Theory of Signal Detectability if P(SN) were known: use that point with slopel with slope1 - P(SN) VN'CA + KN.A = 8 1-P(sN) P(SN) VI 5 P(SX) P(SN) VSN.A + KSNCA P Using this equation, Fig. 1, and the equations for Vo and V1 -(2.3n) and (2.3n) - V can be computed as a function of P(SN) only. This is shown in Fig. 3. The minimum is at approximately P(SN) =.28; V =.40. The receiver 8. 72+ solution can then be obtained by determining, F = 8 = 1.4, and from Fig. 1 the point on the upper bound with slope 1.4 is PN(A) =.20 and PSN (A) =.56. This graphical method of solution is straight forward, but is useful only for particular solutions. Section 2.3 and 2.6 derive general solutions that are easily applied. 4 VNCA _ o.2.4.6.8 1.0 P(SN) FIG 3 THE MAXIMUM VALUE OF THE PAYOFF V FOR KNOWN A PRIORI PROBABILITY P(SN) 1 This equation is introduced in EDG TR-13 in section 1.4, Theorems 1 and 7 prove that it applies to the upper curve, and Theorem 8 proves that B is the slope of the tangent. 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2.3 The General Receiver Solution 2.3.1 Derivation and Proof of Solution. The solution of the game is defined by the min-max relation: if P(SN) = p* and criterion A* are solutions, then V(A*,p*) = min max V = max min V. P(SN) A A P(SN) If we choose to minimize over P(SN) = p first, we observe from (2.5) that for any fixed A the payoff is linear in p and therefore aV/d p is a constant. Since the transmitter wishes to minimize V by choosing the proper p for each A, he will choose p either O or 1 except for those A such that V is independent of P(SN), i.e., unless =. aP A If p* = 0 were the solution, Eq (1.3) would be Vo(A) < V (A*) < V(A*, p) for all p where A* is the corresponding solution for the receiver. The left hand inequality implies that A* = ~, the empty set. If p = 1 is used for the right hand inequality, this becomes Vo(0) < vz(O) which is simply V -K N.CA _-SN.CA contrary to that assumption that V + K > O. A similar contradiction arises if p* = 1 is tried as a solution. Therefore, a solution can be obtained only if c3V p = O. By inspection of (2.5) this is simply the equality of Vo and V1. The above argument can be summarized formally. Theorem 1. A necessary condition that A be an optimum criterion in this game is VSN.A PsN(A) - KSN.CA [1- PSN(A)] = VN.CA [1 - PN(A)]- KN.A PN(A). (2.6a) 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN An equivalent formulation is PSN(A) [VSN.A + SN*CA] +PN(A) [VNCA + KVN.A] VN.cA + SN.CA (2.6b) Theorem 1 gives only a necessary condition for a solution; there may be many criteria for which Eq (2.6) holds. The optimum criterion satisfies this condition with the maximum value of V, that is, with the largest common value V = V1. The problem is to determine the criterion A such that (2.6) holds and that A maximizes PSN(A) [ VSN.A + KSN CA] 1 SCA The assumption that [VsN.A + KSN CAI > 0 simplifies this to the requirement that A maximize PSN(A). Thus the optimum criterion satisfies (2.6) and has maximum PSN(A) with respect to all other others that satisfy (2.6). Because all of the bracketed quantities in (2.6b) are positive, it is convenient to introduce two parameters, al and a2, where a1 = [VN.CA + KNA]/[VSN.A + KSN.CA] (2.7) a2 [VNCA + KSN CA] [vSNA + KSN.A]CA and (2.6b) becomes PSN(A) + a1 PN(A) = a2 (2.6c) The general theory of signal detectability specifies that the optimum receiver for the three general definitions of'optimum" considered in Technical Report EDG-13 can be achieved by a receiver that has as its output the likelihood ratio of its input, and that if the noise is analytic the only way to achieve the optimum is for the output to be the likelihood ratio of the input, or some strictly monotone function of the likelihood ratio. For each input x(t) the likelihood ratio is defined as the ratio of the probability density functions

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN l(x) = fsN(x) / f ) (2.8) and the operator will say that there is signal plus noise present whenever the output is above a predetermined operating level 3. Because the definition of "optimum" is slightly different in this game case from those considered previously the theory cannot immediately be applied to show that such a receiver is the optimum for the game. However, such a receiver is worthy of special consideration as a trial solution, and criteria associated with it are given the special notation A(F), where 0 is the operating level. In a case where the noise is analytic, the set of points with likelihood ratio exactly equal to the boundary value 0 is a set of probability zero, and therefore of no consequence. A more general consideration than analytic noise may require consideration of the boundary. For this reason any criterion A which is like the set A(P) but includes either all, part or none of the boundary, is called an Al(A). Obviously every A(F) is an A1(P) but not conversely. The solution to the game is found by proving the following statement. Theorem 2 If there is a value of such that some criterion A is both an A() and satisfies (2.6), then A is a receiver solution. This is seen if any other criterion B that satisfies the necessary condition (2.6) is compared with A. Split both A and B into their common part and the remainders:l B = (B n A) U (B - A) A = (B n A) U (A - B) For the common part and for A-B, since both are subsets of A, I (x) > P; in B - A, I(x) < I. From (2.6c) then we have 2 = PsN(BnA) + PSN(B-A) + al PN(BNA) + alPN(B- A) 1 CnD is the common part of criteria C and D. CUD is the criterion consisting of both C and D. 15

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN a2 PSN(B NA) + PSN(A B) + a1 PN(BA) + alPN(A -B). Equating and canceling the effect of the common part, PSN(BA) + a1 PN(B -A) = PSN(A -B) + PN(A -B) (2.9) In integral form this is [fsN(X) + 1 fN(x)] dx X [fsN(x) +1 N ] dx, B - A - B and factoring out fN(x) in the integrand fN(x) f1(x) +a1 dx fN(x) [{(x) + 1] dx B -A A -B Now on the left, l(x) + aC1< + aC', and on the right l(x) + al>~ + al. Therefore substitute D + a1 for l(x) + al on both sides (and then cancel this constant factor); the left side dominates the right. /fN(x) dx > fN(x) dx B -A A-B These integrals are the false alarm probabilities on the difference sets B -A and A - B. P (B - A) > PN(A - B). (2.10) This result and equation (2.9) yield PsN(B A) PS (A - B) PSN SN and the proof is completed by adding PSN(BnA) to both sides to obtain the inequality PSN(B) S PSN(A) (2.11)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Thus the receiver solution is the A1(B) that satisfies (2.6), and no other criterion satisfying (2.6) has a larger value of PSN(A), and therefore no other criterion satisfying Theorem 1 has a larger payoff V. Since A1(p) also lead to the upper payoff surface, the argument of Section 2.2 is now completely justified. 2.3.2 Existence. If it could be shown that for all possible pairs of positive numbers (a1, a2) the hypothesis of Theorem 2 is met then the solution would be complete. We can restrict our attention to O 5 a2 < 1 + a (2.12) by definition of al and a2. Consider: L(P) = PSN(A(p)) + a1 PN(A(I)) (2.13) If P1< P2 then L(1i) ~ L(P2). This is true because all of the points with l(x) > P2 have l(x) > 1, and additional points do not decrease L. If there is no P such that L(}) = a21, (this is Eq (2.6c)), there will be some value of P, call it A*, such that L(P*) > a2 and for all larger values L(P) < a2. This situation is sketched in Fig. 4. The drop in L(0) is due to those points with likelihood ratio equal to P*. Since for any P > P*, the criterion A(P) is so small that L() < a2 we use an A1(P) such that L(P*) = a2, that is, since L(P*) > a2, we remove Just enough points of likelihood ratio P* from A(P*). We can certainly do this whenever probability density functions exist.2 Thus, if there is no P such that an A(5) satisfies (2.6c), there is a P such that a slight modification does. 1 This trouble cannot arise if the noise is analytic. See EDG Technical Report No. 13, Appendix B. See EDG Technical Report No. 13, Part I, Lemma 4, p. 40. 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN L(:) + a, L(,* _)_ FIG 4 EXAMPLE OF DISCONTINUOUS L(R) 2-3_3 Summary. The receiver solution specifies that the receiver should have as its output the function (x), the likelihood ratio of its input x. If we compare this game with the first type of optimum in The Theory of Signal Detectability which arose from the same payoff but for known a priori probability P(SN), we have a good example of how game-theory and probability can complement each other in the treatment of a problem. When P(SN) is known, the payoff is maximized by using the bias level = 1 -P(SN) VN.CA KA = 1 - P(SN) a 2.14 P(SN) VSN.A + KSN.CA P(SN) Obviously, knowledge of P(SN) is paramount. Lacking that knowledge, a best "safe" value of v is yielded by the game: a such that PsN(A1(~)) + acl PN(A1(B)) = a2 where al and a2 are defined by Eq (2.7).

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It is "safe" because the payoff is made independent of the value of P(SN), and it is the best because it yields the maximum payoff over all other safe values. For any different criterion the opponent can lower the payoff below this "best safe" value. However, if P(SN) is known, playing the game solution will not be better than the solution given by (2.14) and will be worse unless the i's luckily coincide. Thus, the receiver is always designed to transform each input x = x(t) into the output l(x). Then either probability or game solutions determine the level f depending on knowledge or lack of knowledge of P(SN). 2.3.4 Use of Receiver OPerating Characteristics Curves to Obtain Particular Solutions. For those who are not familiar with the notion of a receiver operating characteristic the following may help to explain it and its usefulness. The ROC curve is a graph of PSN(A) vs PN(A) for all A1(P) criteria. It is often drawn together with the graph of its slope. A typical ROC curve is (1) in Fig. 5a; the diagonal marked (2) represents the effect of ignoring the receiver and guessing; the curve (3) is simply the first curve rotated about the (.5,.5) point and is useful in theory because the area between (1) and (3) inclusive contains all of the possible combination of PSN(A), PN(A) for any particular situation (i. e., fixed types of signals and noise and their energies, fixed observation time, etc.) and therefore any receiver of any type is represent ed by a curve in this area. Equation (2.6c) can be plotted on the R.O.C.; it is a straight line from (O, QA) with slope a1' Thus all of the points on this straight line and between or on curves (1) and (3) represent possible receivers that satisfy the necessary condition PSN(A) + a1 PN(A) = a2. (2.6c) 19

U., 1.0 1.0z 0 a2 a2 oel2)~ o %.- 0 ~~~~~~~~~~~~~~.6..4' SOLUTION TO RECEIVER 2 (3)-!2~-CC.2 0~~~~~~~~~~~~~~~~~ o ro 0.2.4.6.8 1.0 0.2 1S4.6.8 1.0 0N(A) PN(A) -' ~ ~ ~ ~ ~ ~ N 10 10 ~7 711 r r ~~~~~ ~~I Q 5 - 5 o1 PN(A IPNA 3 W _ — ~I J W 3 w ~ ~ ~ ~ ~ ~~~~~~~wI 2 - _~,r1 - 2 z I I~~~~~~~~~~~ Q Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r w~- Iw "~0.7 O. 0.-.- _.55 0.3 2131 111..I 0. - FIG 5a FIG 5b TYPICAL RECEIVER OPERATING TYPICAL RECEIVER OPERATING CHARACTERISTIC CHARACTERISTIC WITH GAME SOLUTION FOR al=.5, Og.9

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Obviously the intersection with curve (1) yields a maximum value of P (A) SN among these. For example, if the ROC is as in Fig. 5, and if a2'= *9, a =.5, the solution is obtained in Fig. 5b. From that graph we read PN(A) =.33, PSN(A) = 74, and A = A(.55), i.e., the optimum receiver which generates likelihood ratio should be set at an output bias level of.55. 2.4 The General Transmitter Solution 2.4.1 A Point Solution. A solution can be obtained for the transmitter if we maximize first over A and then minimize with respect to P(SN). The maximization restricts the receiver so that PSN(A) and PN(A) lie along the receiver operating characteristic with the operating level a function of P(SN), Eq (2.14). Further, if PN(A) is considered as the independent variable, the slope of the curve at any point is the value of the operating level associated with that point. Call the solution for the receiver operating level US. The existence of a min-max solution requires that this PS maximize V for P at his solution, and therefore the derivative of V with respect to the receiver will be zero at the solution. Recalling V = P(SN) V1 + (1P(S o)), (2.5) o N.CA -[NCA + KN.A]PN(A) (2.3) v1 [VSNeA + KSN.A] PSN(A) - KSNA' (2.4) we form PNA) P(SN) [ VSNA + KSN.A] - (1 - P(SN)) [ VN CA+ N.A] (2.15) and letting F and VN.CA + KN.A (2.16) [VSN.A + KSN.A] %s +[VN'CA + KN.A].S, l 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Note that if (2.16) is solved for PS that ~s - z - P/,SN) S p(sN) 1 checking the fact that (2.16) yields an equilibrium point, for if the transmitter P chooses P(SN) = 1l and tells the receiver, the receiver will not Os + al change from the point at which he is already operating (based on no knowledge of P(SN)). 2.4.2 Uniqueness of the Receiver Solution. In section 2.2 a solution to a game with payoff V(p) was defined as an equilibrium point (a*, p*) such that Eq (1.3) held. V(a, p*) < V(a*, p*) < V(a*, p) (1.3) This implies that the solution shall consist of a single criterion A and a constant value of P(SN), and such solutions were obtained. Actually this is only one manner of playing the game. It is the simplest manner of play, and V* = V(A(PS), )+ S) is the unique value of the game, since the receiver can actually attain a payoff V* by using A(%S) while the transmitter can act to hold the value down to this level by letting P(SN) 1. There may however be equally advantageous manners of playing of a more complex type; namely, where the players choose probability distributions for their respective variables instead of simple solutions. The modified form of Eq (1.3) is then V(F, G*) < V(F*G*) < V(F*, G) (2.17) where V(F, G) = J V(a, p) dF(a) dG(p) and F*, G* are optimum distributions. This is nothing more than averaging over the various combinations that occur. It is assumed that the two variables are 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN independent, that is, although whether the receiver operator decides to say "yes, there is signal plus noise" or not is dependent on whether a signal was actually present or not, the criterion for deciding is independent of the instantaneous (unknown to the receiver) a priori probability of these being a signal present. One optimum distribution G* for the transmitter is that already obtained: to let P(SN) be exactly 1. Eq (2.17) then becomes fv (a, C dF(a) < V* < v(a,p) dF*(a) dG(p) (2.18) Letting G(p) = G*(p) and restricting our attention to Ai(d) type criterion, the right hand inequality becomes a necessary condition for any candidate for an optimal distribution H*(f) for the receiver Jv (),... d*() (2.19) PS+ "1 If the value P(SN) = is used in Eq (2.15)for the derivative of V with PSali respect to the false alarm rate the result simplifies to (v A Sa1) = [VN-CA + KN*A [VSNA + KSN.CA] dPN(A) [VSN.A + KSN.cA] [+ VNCA + KN.A] Because calCa1 av( 1( )'sS:' )(Av( ), +al d PN(A1(P)) da daPN(Al() ) d) 23

ENGINEERING RESEARCH INSTITUTE. UNIVERSITY OF MICHIGAN awhere the second factor is always negative, PN(A ill have the same sign as (SS-f), and V has a unique maximum of 0 = fS. This means, of course, that V* > v (Al(), -i for i / S (2.21) V* = V (Al(), — 1) for a = PS The only way that (2.19) can be satisfied is for H*(P) to be a step function at =,S that is, for the receiver to use the criterion Al(%S) and not vary from that in any manner. 2.4.3 Distribution Solutions, Non-Uniqueness of Transmitter Solution. Let us now allow the transmitter player to use a distribution for the a priori probability. Because both the value of the game V* and the receivers unique solution are known, the defining equation of a solution, Eq (2.17) is greatly simplified. The right hand inequality is automatically satisfied by the receiver solution and the left hand inequality becomes ff V(A1(P), p) dG*(p)dH(P)< V* (2.22) where G*(p) is an optimum solution for the transmitter and H(P) is any distribution for the receiver (not necessarily optimum). This can be simplified to requiring that JV (A1(P), p) dG*(p) < V* for all P (2.23) because if (2.23) holds so does (2.22); and if (2.23) does not hold, say for 6 = A*, then H(P) can be a step function at = A* and (2.22) would not hold. Because for fixed n the value V is a linear combination of VO and V1, V(A1(6), p) = PVl(Al(6)) + (1 - p) Vo(Al(6)) 24

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Eq (2.23) became l1(Al(w)) [ pdG*(p) + V (Al()) [1 - pdG*(p)] < V* for all P (2.24) In the previous section it was shown that v ()'1 <* for all (2.21) That is, Vj(A 1 S?(al + Vo(A1(w) - PS?(-l ]< V* for all Thus any distribution G*(p) with the correct mean value J pdG*(p) = Sa(2.25) will satisfy the necessary and sufficient condition (2.23) and therefore (2.22) and is therefore an optimum solution equivalent to P(SN),= 1 (2.26) 2.5 The General Solutions Applied to the Numerical Illustration Let us assume that the nature of the signal and noise are such that the receiver operating characteristic is Fig. 6, e.g., d = 1. This particular operating characteristic occurs repeatedly in detectability. For payoff values, let VsN.A = $7 -KSN.CA = $8 -KN-A = -$6 VN.CA = $2 per unit time T. These are the values used in constructing the model, Fig. 2.

1.0,,_ i I / 9./ _ " I ___.__,f _ _ / ol d 81lll E un S E du 16 ~~~~~~~~0.9<10ZIIIL 1-0VL1 I 1001 1 0.9 1 1 l I I/ o d -=4 - 1 P 07 X tYlI II1 dIb5 / jr.. 41 / " i, j.H 0.i ff / I I I' I I I I' I I II I I I I~~~~"00 I woe / one~ dt I l I Y I I1/ _JI I' I__ 0.7'too 0.6 z 0.5 Ur~~~~- OF OF r / I 0.4 I 1 11 1/1 A IJI/I IL _1 I IOle I! I 0.3 0.2 0.1 i 0. 1 0.1! 04 1 Q6 0.7 1, 0.9 I' 0.5111 1 Is 1rl 1 1,I I IK i- l',Zl' I I I I I / ~_________ -o [11 I!l I I V/ _! IA I 1 ii 1X~ 1/ ______________|I|i~t 1P 1/ll/ I F I-/11 1/I_ I I.t'EFl 1 11____________-I I I -1 04 I 1/1 I/11 s I I I 1 1 ______________ II III IIII:I.4i l I -12 Il I _%1TZ I I A I I TI I I -_ I I i I, I i FNce I IVlIYI _ IFIG, 6 RECEIVER OPERATING CHARACTER~ISTIC. 4L fH iJ]/ / L I F, = [' F = ll Ia ~ I]T]LI~EITE IT,r. P ~MI-MNP {I Ul l!E 1i 1_ I- _ II I I I I I I I _. __ IIIIII I I-iI i l/E1 /1 I!" 1 I1 11 I! Il I!i I I I I IIII."'IeII: IW4I Il/ i/ I/ I I i 1 1!/_ I I I 11111__ I I I I I i I i I I I I 1'1"11 1 1?1/V I/I I_ I I, ~1 1 1 1~ 1 1 I I I'",III L 111 111 o I. l _I _I _ _ _ v J II11 1 I I 1I 1: 1111 < 1 1 1 11!1 11 1 - - 0. II1'1 11[ 1 _III I. _ I I i 111111___ i I i I I I I I I I! I I I II I I I I O 111!/11 I _1T1 1111 I_ I I! I'1- 1F ITI 11! iT 0 0.1 0.2 0.3 0.4 05 Q6 0.7 GS 0.9 1.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, I~~~~~~~~~F i,/ ~,,.,FIG! REEVROEAIG/HRCEITC WITH GAME SOLUTION FOR al~~~.53, a2-.67 O2 SANRA EIT IT r2zeS2(S-N2=d,,' 0.1 02 0. 0,4,5 06 0 7 Q..

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If the receiver were dead, the best possible thing for the operator to do is given by the usual game solution for independent processes found by the intersection of (2.6c) and the diagonal, and P(SN) = C1/(l + al): P(SN) =.348 P(A) =.435 V = -2.478 and the best the receiver operator can do against a smart transmitter is to say there is signal plus noise 43.5 per cent of the time and to lose $2.48 per unit time. However, with a receiver operating, the transmitter still uses a random process, but the receiver player is no longer independent of the transmitter. To obtain the receiver solution, compute 2+8 2/ a2 2+8 a2 += 2/3 - = + =1.25 a= 8/15 Plotting (2.6c) on the ROC we can obtain the solutions from the intersection of (2.6c) and the curve d = 1, and P(SN) =. Using the interPS + a1 section of (2.6c) and d = 1 yields PSN(A) =.56 PN(A) =.20 and the slope of the curve gives the associated PS = 1.40. From these and P(SN) = al/BS + al P(SN) =.533 =.27586, V =.40 1.4 +.533 Thus operating the receiver has forced a reduction of P(SN) from 35 per cent to 28 per cent and increased the payoff from a cost of $2.48 per unit time to a gain of $.40 per unit time. This change has been accomplished with very poor reception. 27

ENGINEERING RESEARCH -INSTITUTE * UNIVERSITY OF MICHIGAN In a similar fashion, if the reception is characterized by d = 4, the intersection of d = 4 and (2.6c) yields P (A).64 PN(A) =.o56 1 = 3 SN N and therefore P(SN) =.1818 V = 1.55 2.6 Limiting Cases as Detection Improves It might appear from the example that as detection improves (due to a decrease of noise or increased signal energy, observation time or other reason) that false alarm probability and a priori probability both go to zero. Actually, this is always the case when VSN.A > VN.CA. The solution to the receiver is obtained by the intersection of VO(Al(p)) and Vl(Al(P)). Now VO is a linear function of PN(A) decreasing from VN.CA to -N A independent of the perfection of the detection. On the other hand, V1 increases monotonicly from - CA to V and as detection improves the rise SN.CA SN'A becomes sharper. If VSN.A > VN.CA, the intersection will occur at a value slightly less than that of the smaller, VN.CA, and therefore for a very small value of PN(A), and thus large values of P. The level P normally isunbounded, and therefore as l becomes larger and larger, the a priori probability P(SN) will approach zero. If VSNA VN.CA, the intersection will occur when Vo is slightly smaller than VSN.A. Using the linearity of Vo the PN(A) at the solution is;ion slightly greater than PN(A) VSN.A- VN CA VN.CA + -A 1 See examples in Technical Report, EDG No. 13, Part II. All those considered require -.OO for PN(A)- O for any degree of detection. 28~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For any fixed false alarm rate, as detection improves, the operating level B normally approaches zerol and therefore P(SN) approaches one. Obviously VSN.A = VN.CA must be treated separately and more carefully because slightly to either side P(SN) goes to zero or to one. One special case of interest is that of Fig. 4.1 of EDG Technical Report No. 13 in which the ROC curves are symmetric. A straight line PSN(A) + PN(A) = 1 intersects all of the ROC curves at 0 = 1 while a straight line PSN(A) + a1 PN(A) = 1 intersects either increasingly larger or decreasingly smaller values of P depending on whether al is greater than or less than one. The limiting cases are summarized in Table I. CASE 3S P(SN) PsN(A) P (A) V VCA+ KSN-A VSN >V a < O O NCA SN A o V SN-A N-CA' 2 V +-KSA N-CA VSN-A >V-c a2SNA A N-CA VSN'A = VN'CA X a2'? IO V'SNA -VN CA t ni NORMAL 0 0 o KSN <K N - SN.CA N-A g'n' NORMAL o 2 K SNCA >K N-A KSNvS-CA NCA - <V a 1CA VSNA VSNCA SN A TABLE I. LIMITING VALUES AS DETECTION IMPROVES. 1See examples in Tech. Report. EDG No. 13, Part II, Fig. 4.5 where I — S >0 as PN(A) — 1 for any realizable detection, but aSsO as detection improves. 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3. SOLUTIONS WHEN INFORMATION IS AVAILABLE There may be instances in which the receiver operator has some information about the a priori probability but not complete knowledge of the value of P(SN). 3.1 A Priori Probability Restricted to a Known Interval In Section 2 it was assumed that P(SN) could take on all values between zero and one. This may very well not be the case; in fact, the range of P(SN) may be very narrow. Therefore let us assume that O< p < P(SN)<p <_ (3.1) o 1 and a game-type solution is desired. The solution to this is readily obtained. When po = 0 and P1 = 1 we used the fact that neither V(po) nor V(pl) dominated the other for all criteria, and therefore the solution was interior, i.e., was given by a criterion A that was neither the empty set nor the set of all inputs, and by a priori probability that was neither po = 0 nor P1 = 1. Now if the game solution for the transmitter for the case when P(SN) was allowed to range from zero to one is within the limits of interest o<. 1+ 1l < P1 (3.2) then the solutions are precisely the same to both receiver and transmitter, as in Section 2. This is apparent when we observe that at the solution in Section 2 the min-max relation held for all a priori probabilities and all criteria, and therefore it still holds for restricted values as long as the solution is among these restricted values. This applies to both types of transmitter solution.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If the game solution for P(SN) falls outside of the range of interest, that solution can still be used to obtain a solution within the range of interest First, we observe that at all times the receiver player will use a likelihood ratio type receiver, because the solution must be stable, and whatever the solution value p of the a priori probability the receiver solution will satisfy (2.14) that is, be a likelihood ratio type receiver with $ = - al. Second, we observe that there was a unique value of P such that VO = V1, namely, the game solution value given in Section 2.3 and correspondingly a unique value for the solution for the a priori probability. Thus for only this value of P(SN) does the corresponding 0 yield Vo = V1. Because the payoff surface is continuous, in any restricted range of a priori probability not containing this unique solution either Vo dominates V1 throughout, or V1 dominates Vo together. Consider an interval such as O 0 p < pl<. To determine S + al whether Vo dominates V1 or vice versa, we can try at any point, for example, p = O. If p = 0, the corresponding optimum criterion is the empty set 0, and we have previously seen that for the empty set V dominates V1. Thus for any + a1 Vf'A(L. "\dominates V a1 ( 1 a minate VA 1 1( -p a1)). Recalling (2.5) PS + alc o p 1' V(A) = p V1(A) + (1 - p) Vo(A) (2.5) and using the dominance Vo(A1 (a)) - V1(Al(P))> 0 we can rewrite V as V(A1(P)) = Vo(A )) - p [ V(A1() - V1(A1()] (3-3) for all P associated with p< 1. Therefore if the range of a priori probS + a1 ability is restricted to po < p - P1 <1 (3.4) 31

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN by using the min-max principle we can consider A as fixed and minimize first with respect to p, which in all cases will result in the choice p = P1 (3.5) and then maximizing over A yields correspondingly 1 - Pl: = P1 al (3.6) Conversely if aI US < Po< P <- P1 (3.7) the dual argument yields p = p (3.8) and correspondingly P - p = - ~" 1 (3.9) Po Summarizing the arguments, since either order of maximization and minimization leads to the same result, we observe that maximizing first with respect to A would cause the receiver to be of the likelihood ratio type operating on a portion of the ROC where VO dominated V1 throughout (or vice versa). If the minimization over p is carried out first, for all those criteria on the range of the ROC indicated above the same extreme value P1 (or po) is chosen and therefore is independent of the choice of A in the subsequent maximization. 3.2 A Priori Probability a Random Variable With Known Mean It was shown in Section 2.4.3 that any distribution of a priori probability with the proper mean was a transmitter solution, and correspondingly the receiver has no choice but to use A (5S), the unique solution, as criteria. 1 It may occur that the mean is known to be p. If p is. then knowing this 32

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN is of no value to the receiver. This is why the game solution (to the transmitter) is called "equilibrium". Suppose that p is not this equilibrium value. The receiver wishes to maximize V (H(P), G(p)), the average payoff. V(H, G) = V(A1( ), P) dG(p)dH(P) for a fixed P, as in Section 2.4.3 JV(A1(P),p) dG(p) = Vl(Al(A) fpdG(p) + Vo(Al(P)) - Vo(A1(w)) pdG(p):p Vl(Al(o)) + (1 - p) Vo(B)) Because this is maximized as in Eq (2.14) by i = 1 - al (3.10) p the average payoff V(H,G) will be maximized by the trivial distribution H that has a jump at P — i.e., the receiver should use an A1( P al) criterion. p Although this is what would be expected, it is important in that it points up the fact that the receiver operator is unable to make use of any information about the transmitters distribution other than its mean, in order to maximize the average payoff V(H,G). 4. SUMMARY OF SOLUTIONS The optimum receiver for the cases considered is, as in EDG Technical Report No. 13, that receiver that has as its output the likelihood ratio function of its input. 1) If the a priori probability may range from zero to one, the operating level S of such a receiver (that is, the bias level such that if the output 33

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN exceeds the bias, it is profitable to say there was a signal present) is adjusted so that PsN(Al(PS)) + alPN(Al(pS)) = a2 (2.6c) where l and a2 are constants depending onthe various values placed on the possible responses, Eq (2.7). The solution for a priori probability is P(SN) = a1 aS + a1 (2.15) The receiver operating characteristic, a plot of PSN(A) vs PN(A), is a convenient device for solving for both PS and P(SN). 2) If the a priori probability is restricted to range from Po to P1 inclusive, the transmitter solution is given by (2.15) if that value is between po and P1, or is as close that value as possible, and the receiver solution is accordingly P(SN) Ia1 (2.14) 3) If the a priori probability is a random variable with known mean value, then the receiver solution is a = 1 - P al~ (3.10) P As a consequence of the min-max principle, the optimum value of p is + a1 S + al Any distribution of a priori probability with this mean value is an optimum distribution from the anti-detection viewpoint. This allows the transmitter considerable freedom to achieve other purposes. 34

LIST OF SYMBOLS A The event "The operator says there is signal plus noise present," or a criterion, i.e., the set of receiver inputs for which the operator says there is a signal present. A(P) The criterion of all receiver inputs with likelihood ratio not less than A. A1(B) Any criterion differing from A(P) only on receiver inputs with likelihood ratio equal to B. Af B The common part of criteria A and B. A - B All receiver inputs in A but not in B. CA The event "The operator says there is noise alone." fN(X) The probability density for points x in R if there is noise alone. f N(x) The probability density for points x in R if there is signal plus noise. F(a), F*(a) A probability distribution function associated with the receiver's strategy. G(p), G*(p) A probability distribution function associated with the transmitter's strategy. H(P), H*(P) A probability distribution function associated with a likelihood ratio receivers strategy. iK*CA Value of the error of falsely saying a signal was present. KSNCA Value of the error of missing a signal. KSN.CA fSN(X) I(x) The likelihood ratio for the receiver input x. l(x) = sN fN(x) L(P) L(F) = PsN(A(p)) + a61 PN(A(p)) N The event "there is noise alone." P(SN), p A priori probability of a signal being transmitted. Po' P1 Bounds on p. p The average value of p. PN(A) The probability that the operator will say there is signal plus noise if there is noise alone, i.e., the false alarm probability. 35

PSN (A) The probability that the operator will say there is signal plus noise if there is signal plus noise, i.e., the probability of detection. R The space of all receiver inputs; as a criterion R indicates the receiver always indicates a signal is present. SN The event "There is signal plus noise." T The unit of time of transmission; the duration of the observation. V, V(ap) The payoff function for fixed strategies, Vo(A) The payoff when p = 0. V1(A) The payoff when p = 1. V(F,G) The payoff function for distributed strategies. V* The value of the payoff function at the solution. V SN.A Value of correctly detecting a signal's presence. VN.CA Value of correctly detecting that no signal was present. x A receiver input x(t). 1 1 = [VN'CA + K.A] [VSN.A + KSN.CA a2 a2 = [VNCA + KSN. CA] [SNA + KSN.CA] 3,S The optimum operating level of a likelihood ratio receiver. The empty set; as a criterion 0 indicates the receiver never indicates a signal is present. Note: An * usually indicates optimum or solution functions or values. 36

DISTRIBUTION LIST Copy No. 1 Director, Electronic Research Laboratory Stanford University Stanford, California Attn: Dean Fred Terman Copy No. 2 Commanding Officer Signal Corps Electronic Warfare Center Fort Monmouth, New Jersey Copy No. 3 Chief, Engineering and Technical Division Office of the Chief Signal Officer Department of the Army Washington 25, D. C. Attn: SIGGE-C Copy No. 4 Chief, Plans and Operations Division Office of the Chief Signal Officer Washington 25, D. C. Attn: SIGOP-5 Copy No. 5 Countermeasures Laboratory Gilfillan Brothers, Inc. 1815 Venice Blvd. Los Angeles 6, California Copy No. 6 Commanding Officer White Sands Signal Corps Agency White Sands Proving Ground Las Cruces, New Mexico Attn: S IGWS -CM Copy No. 7 Signal Corps Resident Engineer Electronic Defense Laboratory P. 0. Box 205 Mountain View, California Attn: F. W. Morris, Jr. Copy Nos. 8-82 Transportation Officer, SCEL Evans Signal Laboratory Building No. 42, Belmar, New Jersey For - Signal Property Officer Inspect at Destination File No. 25052-PHI-51-91(1443)

UNIVERSITY OF MICHIGAN 1IIII11111111111111 II III I 3 9015 02229 2497 Copy No. 83 W. G. Dow, Professor Dept. of Electrical Engineering University of Michigan Ann Arbor, Michigan Copy No. 84 H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan Copy No. 85 Document Room Willow Run Research Center University of Michigan Willow Run, Michigan Copy Nos. 86-95 Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan Copy No. 96 Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan'38