Static two-player evasion-interrogation games with uncertainty

Static two-player evasion-interrogation games with uncertainty^*

H.T. Banks; S.L. Grove; K. Ito; J.A. Toivanen

Center for Research in Scientific Computation, North Carolina State University Raleigh, NC 27695-8205. E-mails: [e-mail]/ [e-mail]/ [e-mail]/ [e-mail]

ABSTRACT

We consider electromagnetic evasion-interrogation games wherein the evader can use ferroelectric material coatings to attempt to avoid detection while the interrogator can manipulate the interrogating frequencies (wave numbers) and angles of incidence of the interrogating inputs to enhance detection and identification. The resulting problems are formulated as two player games in which one player wishes to minimize the reflected signal while the other wishes to maximize it. Simple deterministic strategies are easily defeated and hence the players must introduce uncertainty to disguise their intentions and confuse their opponent. Mathematically, the resulting game is carried out over spaces of probability measures which in many cases are appropriately metrized using the Prohorov metric.

Mathematical subject classification: 49N75, 91A05, 91A10, 91A60, 78A30, 78A46.

Key words: pursuit-evasion games with uncertainty, electromagnetic interrogation, inverse scattering.

1 Introduction

In [8] the authors demonstrated that it is possible to design ferroelectric materials with appropriate dielectric permittivity and magnetic permeability to significantly attenuate reflections of electromagnetic interrogation signals from highly conductive targets such as airfoils and missiles. This was done under assumptions that the interrogating input signal is uniformly likely to come from a sector of interrogating angles a Î [a₀,a₁] (a = - f where f is the angle of incidence of the input signal) but that the evader has knowledge of the interrogator's input frequency or frequencies (denoted in the presentations below as the interrogator design frequencies I_D). These results were further sharpened and illustrated in [9] where a series of different material designs were considered to minimize over a given set of input design frequencies I_D the maximum reflected field from input signals. In addition, a second critical finding was obtained in that it was shown that if the evader employed a simple counter interrogation design based on a fixed set (assumed known) of interrogating frequencies I_D, then by a rather simple counter-counter interrogation strategy (use an interrogating frequency little more than 10 % different from the assumed design frequencies), the interrogator can easily defeat the evader's material coatings counter interrogation strategy to obtain strong reflected signals.

From the combined results of [8, 9] it is thus rather easily concluded that the evader and the interrogator must each try to confuse the other by introducing significant uncertainty in their design and interrogating strategies, respectively. This concept, which we refer to as mixed strategies in recognition of previous contributions to the literature on games (von Neumann's finite mixed strategies to be explained below), leads to two player non-cooperative games with probabilistic strategy formulations. These can be mathematically formulated as two sided optimization problems over spaces of probability measures, i.e., minmax games over sets of probability measures. The purpose of this paper is to provide a mathematically precise formulation of such a class of two sided optimization problems and to discuss our initial computational efforts on such problems. Approximation methods are introduced and their computational efficacy is demonstrated with several simple examples.

2 Problem formulation

We consider electromagnetic interrogation of objects in the context of minmax evader-interrogator games where each player has uncertain information about the adversary's capabilities. The minmax cost functional is based on reflected fields from an object such as an airfoil or missile and can be computed in one of several ways [8, 9].

The simplest is the reflection coefficient based on a simple planar geometry (e.g., see Figure 1) using Fresnel's formula for a perfectly conducting half plane which has a coating layer of thickness d with dielectric permittivity e and magnetic permeability µ. A normally incident (f = 0) electromagnetic wave with the frequency f is assumed to impinge the half plane. Then the corresponding wavelength l in air is l = c/f, where the speed of light is c = 0.3 ×10⁹. The reflection coefficient R for the wave is given by

where

This expression can be derived directly from Maxwell's equation by considering the ratio of reflected to incident wave for example in the case of parallel polarized (TE_x) incident wave (see [8, 20]).

An alternative and much more computationally intensive approach (which may be necessitated by some target geometries) employs the far field pattern for reflected waves computed directly using Maxwell's equations. In two dimensions, for a reflecting body W with coating layer W₁ and computational domain P with an interrogating plane wave E⁽ⁱ⁾, the scattered field E^(s) satisfies the Helmholtz equation [15]

where the Silver-Müller radiation condition has been approximated by a second-order absorbing boundary condition on ¶P as described in [2, 17, 18]. The vectors n and s denote the normal and tangential directions on the boundary ¶P, respectively, and C is the set of the corner points of P. Here [ · ] denotes the jumps at interfaces. The incident field in air is given by

E⁽ⁱ⁾ =

where a is the interrogation angle (a = - f) and k = 2pf/c = w/c is the interrogation wave number corresponding to the interrogating frequency f. The corresponding far field pattern is given by [8, 15]

where E^(s)(x₁,x₂;e,m,a,f) is the scattered electromagnetic field. This can be used as a measure of the reflected field intensity instead of the reflection coefficient R of (1)-(2).

The evader and the interrogator are each subject to uncertainties as to the actions of the other. The evader wants to choose a best coating design (i.e., best e's and µ's ) while the interrogator wants to choose best angles of interrogation a and interrogating frequencies f. Each player must act in the presence of incomplete information about the other's action. Partial information regarding capabilities and tendencies of the adversary can be embodied in probability distributions for the choices to be made. That is, we may formalize this by assuming the evader may choose (with an as yet to be determined set of probabilities) dielectric permittivity and magnetic permeability parameters (e,m) from admissible sets E×M while the interrogator chooses angles of interrogation and interrogating frequencies (a,f) from sets A×F. The formulation here is based on the mixed strategies proposals of von Neumann [1, 22, 23] and the ideas can be summarized as follows. The evader does not choose a single coating, but rather has a set of possibilities available for choice. He only chooses the probabilities with which he will employ the materials on a target. This, in effect, disguises his intentions from his adversary. By choosing his coatings randomly (according to a best strategy to be determined in, for example, a minmax game), he prevents adversaries from discovering which coating he will use-indeed, even he does not know which coating will be chosen for a given target. The interrogator, in a similar approach, determines best probabilities for choices of frequency and angle in the interrogating signals. Note that such a formulation tacitly assumes that the adversarial relationship persists with multiple attempts at evasion and detection.

The associated minmax problem consists of the evader choosing distributions P_e(e,m) over E×M to minimize the reflected field while the interrogator chooses distributions P_i(a,f) over A×F to maximize the reflected field. If B_R(e,m,a,f) is the chosen measure of reflected field and _e = (E×M) and _i = P(A×F) are the corresponding sets of probability distributions or measures over E×M and A×F, respectively, then the cost functional for the minmax problem can be defined by

The problems thus formulated are special cases of classical static zero-sum two player non-cooperative games [1, 12] where the evader minimizes over P_e Î _e and the interrogator maximizes over P_iÎ _i. In such games one defines upper and lower values for the game by

= J(P_e, P_i) and J =

J(P_e, P_i)

The first represents a security level (worst case scenario) for the evader while the latter is a security level for the interrogator. It is readily argued that J < and if the equality J^* = J = holds, then J^* is called the value of the game. Moreover, if there exist Î P_e and Î P_i such that

then (, ) is a saddle point solution or non-cooperative equilibrium of the game.

To investigate theoretical, computational and approximation issues for these problems, it is necessary to put a topology on the space of probability measures. Given a set I, a natural choice for (I) is the Prohorov metric topology [14, 19, 21] as used for minimization/inverse problems in [3, 4, 7, 11]. Prohorov metric (r) convergence is weak^* convergence in when is considered as a subset of the topological dual (I) of the space C_B(I) of bounded continuous functions on I. More precisely, convergence r(P_k,P) ® 0 in the Prohorov metric is equivalent to

_If(x) d P_k(x) ® _I f(x) d P(x)

for all bounded, uniformly continuous f : I ® ¹.

It is known [3, 4] that if I is a complete metric space, then taken with the Prohorov metric is a complete metric space. Moreover, if I is compact, then so is . Using these properties and arguments similar to those in [3, 6, 7, 10, 11], one can develop well-posedness and approximation results for the minmax problems defined above. Efficient computational methods that correspond to von Neumann's finite mixed strategies [1] can readily be presented in this context. These can be based on several approximation theories that have been recently developed and used. The first, developed in [3] and based on Dirac delta measures, is summarized in the following theorem.

Theorem 2.1. Let I be a complete, separable metric space, B the class of all Borel subsets of I and

(I) the space of probability measures on (I,B). Let I₀ =

be a countable, dense subset of I. Then the set of P Î (I) such that P has finite support in I₀ and rational masses is dense in

(I) in the r metric. That is,

is dense in (I) taken with the r metric, where D_xj is the Dirac measure with atom at x_j.

It is rather easy to use the ideas and results associated with this theorem to develop computationally efficient schemes. Given

(a ''partition'' of I) chosen so that I_d is dense in I, define

Then we find

(i)

^M(I) is a compact subset of (I) in the r metric,

(ii)

^M(I) Ì ^M+1(I) whenever I_M+1 is a refinement of I_M,

(iii) ''

^M(I) ® (I)'' in the r topology; that is, for M sufficiently large, elements in (I) can be approximated in the r metric by elements of ^M.

A second class of approximations, based on linear splines, was developed and used in [11] for problems where one assumes that the probability distributions to be approximated possess densities in L².

Theorem 2.2. Let F be a weakly compact subset of L²(I), I compact and let

_F(I) º {P Î (I): P' = p, p Î F}. Then

_F(I) is compact in

(I) in the r metric. Moreover, if we define

to be the linear splines on I corresponding to the partition I_M, where È_MI_M is dense I, define

and if

º { P_M Î (I) : P_M = p^M, p^M Î ^M}

we have È_M

is dense in _F(I) taken with the r metric.

A study comparing the relative strengths and weaknesses of these two classes of approximation schemes in the context of inverse problems is given in [5].

3 Theoretical results

To establish existence of a saddle point solution for the evasion-interrogation problems formulated above, one can employ a fundamental result of von Neumann [22, 23] as stated by Aubin ([1]-see p. 126).

Theorem 3.1 (von Neumann). Suppose X₀, Y₀ are compact, convex subsets of metric linear spaces X, Y respectively. Further suppose that

(i) for all y Î Y₀, x ® f(x,y) is convex and lower semi-continuous;

(ii) for all x Î X₀, y ® f(x,y) is concave and upper semi-continuous.

Then there exists a saddle point (x^*, y^*) such that

f(x*, y*) =

f(x, y) =

f(x, y).

A straight forward application of these results to the problems under discussion lead immediately to desired well-posedness results.

Theorem 3.2. Suppose E, M, F, F are compact and the spaces X₀ =

(E×M), Y₀ =

(F×F) are taken with the Prohorov metric. Then X₀, Y₀ are compact, convex subsets of X =

(E×M) and Y =

(F×F), respectively. Moreover, there exists (

,

) Î (E×M) ×

(F×F) such that

For computations, we may use the ''delta'' approximations of Theorem 2.1 and obtain

As noted above, a convergence theory can be found in [3]. This corresponds precisely to von Neumann's finite mixed strategies framework for protection by disguising intentions from opponents, i.e., by introducing uncertainty in the players' choices.

To illustrate the computational framework based on the delta measure approximations, we take

which can be represented respectively by

and

We note that in this case the , are the probabilities associated with the use of the material parameters (, ) and interrogating parameters (, ) in the mixed strategies of the evader and interrogator, respectively.

Then J (, ) reduces to

where B_R is a measure of the reflected field (either the reflection coefficient or the far field scattering intensity). Since P^M, Q^N are compact, convex subsets of ^M, ^N, respectively, we have the following theorem.

Theorem 3.3. For fixed M, N there exists (, ) in P^M × Q^N such that

Assume further that (e, m, f, f) ® B_R(e,m, f, f) is continuous on E×M × F×F which is assumed compact. Then there exists a sequence (, ) of elements from P^M×Q^N respectively with corresponding (, ) in (E×M) × (F×F) converging in the Prohorov metric to (, ) which is a saddle point for the original minmax problem.

Proof. The hypothesis that the map (e, m, f, f) ® B_R(e, m, f, f) be continuous on E×M × F×F implies continuity of the map (P_e, P_i) ® J(P_e, P_i ) on X₀ × Y₀® , where X₀ = (E×M) and Y₀ = (F×F) are compact convex in X = (E×M) and Y = (F×F), respectively. This is due to the compactness of E×M and F×F, and properties of the Prohorov metric.

Let P^M, Q^N be defined as in (6), (7), and observe that these are compact convex subsets of ^M, ^N, respectively. Now let P_e Î X₀ and P_i Î Y₀ be arbitrary. Then by Theorem 2.1 and (iii) above (see also [3]), there exists a sequence , Î P^M, Q^N with associated measures , , respectively, such that

® P_e in X₀ as M ® ¥ and ® P_i in Y₀ as N ® ¥

Let , (guaranteed to exist by continuity, compactness, and the von Neumann theorem) with coordinates , , respectively, satisfy

for all (, ) in X₀× Y₀ with coordinate representations in P^M, Q^N, respectively. Now X₀× Y₀ compact implies that there exists a subsequence (, ) and (_e, _i) in X₀ × Y₀ such that (, ) ® (_e, _i) in X₀ × Y₀.

Consider (8) for indices M_k, N_k, so that

J (, ) < J (, ) < J (, )

Then given the continuity of J and taking the limit we find

J (_e, P_i) < J (_e, P_i) < J (_e, P_i).

Since P_e, P_i are arbitrary in X₀, Y₀ respectively, we have (_e, P_i) is a saddle point for the original minmax problem.

Further note that since these arguments hold for any subsequence of (, ), then

J (, ) < J (, ) < J (, )

and ® _e as well as ® _i. That is, any subsequence of , has a convergent subsequence to _e, _i so that the sequence itself must converge to _e, _i.

4 Preliminary numerical calculations

In [9], the authors considered a number of different material designs in their investigation of counter interrogation and counter-counter interrogation feasibility. We use a number of these materials to define families of admissible evader sets E×M in our initial exploratory calculations in demonstrating ease of calculations for problems as outlined above in Theorem 3.3 for the delta measures as defined in (6) and (7). The materials we use here were designed in [9] in the following way. For a given assumed interrogator design frequency set I_D and assumed normal incident waves (incident angle f = 0), the evader designs a material by optimizing over admissible sets for e and µ, the criteria

J_¥(e, m) = B_R(e, m, 0, f)

to obtain ''optimal'' coating parameters (e^*, m^*). With the thickness d = 2.5 millimeters, Fresnel's formula (1)-(2) was used to obtain a series of materials labelled 1R, 2R,...,6R corresponding to (e^*, m^*) = (e_jR, µ_jR) in [9]. Adopting the same labelling here, we allow the evader to choose from subsets composed of elements from the following six materials denoted by their respective parameter values (e_jR, µ_jR), j = 1,...,6:

1R: (e_1R, µ_1R) = (900.4 + 38.2i,1) optimizing only over e with µ = 1 fixed and I_D = {1 GHz}, 2R: (e_2R, µ_2R) = (90.7 + 21.4i, 10 + 2i) for I_D = {1 GHz}, 3R: (e_3R, µ_3R) = (90.7 + 21.4i, 10 + 2i) for I_D = [0.8, 1.25], 4R: (e_4R, µ_4R) = (5625.4 + 95.5i,1) optimizing only over e with µ = 1 fixed and I_D = {0.4 GHz}, 5R: (e_5R, µ_5R) = (558.6 + 29.3i, 10 + 1.2i) for I_D = {0.4 GHz}, 6R: (e_6R, µ_6R) = (549.3 + 150i, 10 + 2i) for I_D = [0.32, 0.5].

Graphs of the resulting reflection coefficients B_R(e_jR, µ_jR, 0, f) = R(e_jR, µ_jR, f) are plotted as a function of possible interrogating frequency in Figure 2. Note that in the cases 1R, 2R, 4R, 5R, if the interrogator uses the design frequency, the evader has a high likelihood of avoiding detection, but his/her evasion strategy is easily defeated by an interrogating signal away from the design frequency.

4.1 Typical results

We carried out a number of different optimization trials using these materials with the ''delta'' measure approximations outlined above. We present in Table 1 a sample of results obtained for several choices of admissible evader material sets E×M and interrogator frequency sets F. The minmax calculations are readily carried out using standard MATLAB minimization packages.

A few words on the findings in the table are in order. First we note that when materials 1R, 2R, 3R (all of which are based on an I_D of 1 GHz or an interval around 1 GHz) are available to the evader, the best strategy for the interrogator is to always choose (with probability 1) the interrogating frequency (.4 GHz) furthermost from the design frequencies. The evader's choice is slightly more delicate. In the first row of Table 1 the evader always chooses material 3R which is based on design for the largest I_D among 1R, 2R, 3R. However, in the second row we see that the evader should use 2R for 24.1% of the time and 3R the other 75.9% of the time. If the materials 4R, 5R, 6R are available, the evader should use material 5R for 66.7% of the time and material 6R the remainder of the time. In this situation the interrogator should use f = .6 GHz if only the frequencies {.4,.6} are available. If {.4,.6,1} are available, the interrogator should use f = .6 GHz for 66.7% of his interrogations and f = 1 GHz for 33.3% of the interrogations to maximize his chances of detecting the evader.

5 Concluding remarks

In this note we have presented preliminary ideas for evader-interrogator strategies in electromagnetic counter and counter-counter interrogation games. While the ideas are simple, they possess potential for readily implementable strategies in a number of other modern non-cooperative adversarial situations in that similar problems arise in information-misinformation dissemination, data retrieval and analysis, and internet traffic and security. The computational results we present are extremely simple, but illustrate aptly the potential. More realistic problems (currently under investigation) involve multiple layers of materials and the corresponding Fresnel's formulae (see [13, 16, 24]) for multiple stacked layers of dielectric materials. And of course one should investigate larger values of M and N than the values M = 3 and N = 2 or 3 used in the preliminary calculations reported on above.

In our computational examples, P_e and P_i were taken as discrete measures with a finite number of atoms, i.e., finite weighted combinations of the delta measures. Our current computational efforts are on absolutely continuous measures dP_e(e, m) = p(e, m)d e d µ and dP_i(f) = q(f)df using the linear spline approximations presented in Theorem 2.2. Specifically, these can be used to obtain approximations

which again reduces considerations to a simple minmax problem for

where now

More generally, one should consider approximations

resulting in a tensor product in the approximating minmax problems with tensor

Of course, in these approximations the p's and q's are no longer simple probabilities (as in the case of the delta basis elements), but are weighting coefficients for the density basis elements , etc., and must be properly constrained so that the finite combinations sum to a probability density.

Acknowledgement. This research was supported in part by the U.S. Air Force Office of Scientific Research under grant AFOSR FA9550-04-1-0220. The authors would like to thank Dr. Richard Albanese of the AFRL, Brooks AFB, San Antonio, TX, for numerous stimulating and valuable comments and suggestions during the course of this effort.

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Received: 01/IX/06. Accepted: 04/X/06.

#688/06.

*

On the Occasion of the 25

^th Anniversary of the Laboratório Nacional de Computação Científica (LNCC), March 21-24, 2006, Rio de Janeiro, Brazil.

Publication Dates

• Publication in this collection
  19 Mar 2007
• Date of issue
  2006

History

• Accepted
  04 Oct 2006
• Received
  01 Sept 2006