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Fictitious Play in Potential Games

This work studies the convergence properties of continuous-time fictitious play in potential games. It is shown that in almost every potential game and for almost every initial condition, fictitious play converges to a pure-strategy Nash equilibrium. We focus our study on the class of regular potential games; i.e., the set of potential games in which all Nash equilibria are regular. As byproducts of the proof of our main result we show that (i) a regular mixed-strategy equilibrium of a potential game can only be reached by a fictitious play process from a set of initial conditions with Lebesgue measure zero, and (ii) in regular potential games, solutions of fictitious play are unique for almost all initial conditions.

Introduction.
A Nash equilibrium (NE) is a solution concept for multi-player games in which no player can unilaterally improve their personal utility. Formally, a Nash equilibrium is defined as a fixed point of the best response mapping-that is, a strategy x * is said to be a NE if where BR denotes the best response mapping (see Section 2 for a formal definition).
A question of fundamental interest is, given the opportunity to interact, how might a group of players adaptively learn to play a NE strategy over time? In response, it is natural to consider the the dynamical system induced by the best response mapping itself: (1)ẋ ∈ BR(x) − x.
By definition, the set of NE coincide with the equilibrium points of these dynamics. Historically, these dynamics are known as fictitious play (FP). 1 After a time change and a discretization, the dynamics (1) yield the familiar (discrete-time) FP process first introduced by Brown in [5] (see, e.g., [4,11,17]).
FP is known to converge to the set of NE in several important classes of games [4,11,23,25,26], though not all games [14,27]. Of particular interest to this work are results showing that FP converges to the set of NE in a class of games known as potential games [3,24,25].
In a potential game, there exists an underlying potential function which all players implicitly seek to maximize. Such games are fundamentally cooperative in nature (all players benefit by maximizing the potential), and have many important applications in engineering and economics [19,24]. Along with so called harmonic games (which are fundamentally adversarial in nature), potential games may be seen as one of the basic building blocks of general N -player games [6].
The set of NE may be subdivided into pure-strategy (deterministic) NE and mixed-strategy (probabilistic) NE. Mixed-strategy NE can be problematic for a number of reasons [14]. In engineering applications involving potential games, mixedstrategy NE can be undesirable since they are nondeterministic, have suboptimal expected utility, and do not always have clear physical meaning [2]. Consequently, in applications, preference is generally given to algorithms that are guaranteed to converge to a pure-strategy NE [2,18,20,21].
For FP learning dynamics, mixed-strategy NE are problematic at an even more fundamental level (both in and out of potential games), since they can cause nonuniqueness of solutions and make it impossible to establish general convergence rate estimates [11].
It has been speculated that, for FP learning dynamics, the difficulties arising due to mixed equilibria rarely occur in practice. For example, in [2] it is noted that "it is generally believed that convergence of [FP] to a mixed (but not pure) Nash equilibrium happens rarely when [players'] utilities are not equivalent to a zero-sum game". 2 Despite such speculation, there are currently no rigorous results showing that FP behaves in this manner (e.g., only reaching mixed equilibria from a null set of initial conditions) in any particular class of games.
An important benefit of potential games is that they guarantee the existence of pure-strategy NE. Nevertheless, it is well known that FP can converge to a mixedstrategy equilibrium in such games. In fact, this deficiency was first noted in the paper where it was originally proven that FP converges to NE in potential games (see [25], Remark (2)).
In this paper, we refine the convergence result for FP in potential games in an attempt to redeem it somewhat in this regard and address the issues noted above. The following theorem is our main result. Theorem 1. In almost all potential games, and for almost all initial conditions, FP converges to a pure-strategy NE.
In particular, we show that in any regular potential game, FP converges to a purestrategy NE from almost every initial condition. The notion of a regular game was introduced by Harsanyi [12]. In a companion paper [28], we show that almost all potential games are regular.
Three important byproducts of the proof of Theorem 1 are (i) in a regular potential game, FP can only reach the set of mixed equilibria in finite time from a null set of initial conditions (see Proposition 15), (ii) in a regular potential game, if FP converges to the set of mixed-strategy equilibria, then it does so in finite time (see Proposition 22), and (iii) except for a null set of initial conditions, solutions of FP are unique in any regular potential game (see Lemma 17 and Remark 21). 3 1.1. Related Work. A related result for two-player games has been shown in [15], where it was demonstrated that FP almost never converges cyclically to a mixed-strategy equilibrium in which both players use more than two pure-strategies. In this paper we focus on N -player potential games and we show that, in such games, convergence to mixed equilibria is generically impossible.
Continuous-time best-response dynamics similar to those we consider here have been studied in various works, including [4,11,13,22]. These papers study a variety of convergence properties for FP and replicator dynamics in a wide class of games, but do not consider the question of generic convergence to pure-strategy equilibria.
As noted earlier, the problem of characterizing the rate of convergence in FP, as studied in [11], is closely related to our work here. The mere existence of mixedstrategy equilibria presents a fundamental barrier to establishing precise convergence rate estimates in FP. This issue is partially resolved by the present work since we show that mixed-strategy NE can only be reached in finite time from a null set of initial conditions (see Proposition 15.) We intend to further address the issue of characterizing the rate of convergence of FP in potential games in a future work.
1.2. Proof Strategy. The basic strategy is to leverage two noteworthy properties satisfied in almost every potential game: 1. The FP vector field cannot concentrate mass in finite time, meaning it cannot map a set of positive (Lebesgue) measure to a set of zero measure in finite time (see Section 5.1).
2. In a neighborhood of an interior Nash equilibrium (i.e., a completely mixed equilibrium), the magnitude of the time derivative of the potential along paths grows linearly in the distance to the Nash equilibrium, while the value of the potential varies only quadratically; that is, where U denotes the potential function and x * is the equilibrium point.
Using Markov's inequality, property 2 immediately implies that if a path converges to an interior NE then it must do so in finite time (see Section 5.2). Hence, properties 1 and 2 together imply that the set of points from which FP converges to an interior NE must have Lebesgue measure zero.
In order to handle mixed NE that are not in the interior of the strategy space (i.e., incompletely mixed equilibria), we consider a projection that maps incompletely mixed equilibria to the interior of the strategy space of a lower dimensional game. Using the techniques described above we are then able to handle completely and incompletely mixed equilibria in a unified manner.
In particular, we see that the set of points from which FP converges to the set of mixed-strategy NE has Lebesgue measure zero. Since any FP path must converge to a NE [4], this implies Theorem 1.
Properties 1 and 2 hold as long the equilibrium x * is regular. In a companion paper we show that, in almost all potential games, all equilibria are regular (see Theorem 4 below).
The remainder of the paper is organized as follows. Section 2 sets up notation. Section 3 discusses regular potential games. Section 4 establishes the two key inequalities used to prove Theorem 1, and Section 5 gives the proof of Theorem 1.

Comparison with Classical
Techniques. Given a classical ODE, one can prove that an equilibrium point may only be reached from a set of measure zero by studying the linearized dynamics at the equilibrium point. Assuming all eigenvalues associated with the linearized system are non-zero, the dimension of the stable manifold (i.e., the set of initial conditions from which the equilibrium can be reached) is equal to the dimension of the stable eigenspace of the linearized system [8]. Hence, to prove that an equilibrium can only be reached from a set of measure zero, it is sufficient to prove that at least one eigenvalue of the linearized system lies in the right half plane.
In FP, the vector field is discontinuous-hence, it is not possible to linearize around an equilibrium point, and such classical techniques cannot be directly applied. However, the gradient field of the potential function may be seen as an approximation of the FP vector field (see Lemma 8). Unlike the FP vector field, the gradient field of the potential function can be linearized. In a non-degenerate game, any completely mixed-strategy NE is a non-degenerate saddle point of the potential function. Hence, at least one eigenvalue of the linearized gradient system must lie in the right-half plane. This implies that, for the gradient dynamics of the potential function, the stable manifold associated with an equilibrium point has dimension at most κ − 1, where κ is the dimension of the strategy space.
Since the FP vector field approximates the gradient field of the potential function, intuition suggests that for FP dynamics, each mixed equilibrium should also admit a similar low-dimensional stable manifold.
While this provides an intuitive explanation for why one might expect Theorem 1 to hold, we did not use any such linearization arguments in the proof of this result. We found that studying the rate of potential production near mixed equilibria (e.g., as discussed in the "proof strategy" section above) led to shorter and simpler proofs. For a finite set S, let (S) denote the set of probability distributions over S. For i = 1, . . . , N , let ∆ i := (Y i ) denote the set of mixed-strategies available to player i. Let ∆ := N i=1 ∆ i denote the set of joint mixed strategies. 4 Let ∆ −i := j∈{1,...,N }\{i} ∆ j . When convenient, given a mixed strategy σ = (σ 1 , . . . , σ N ) ∈ ∆, we use the notation σ −i to denote the tuple (σ j ) j =i Given a mixed strategy σ ∈ ∆, the expected utility of player i is given by For σ −i ∈ ∆ −i , the best response of player i is given by the set-valued function and for σ ∈ ∆ the joint best response is given by the set valued function BR : ∆ ⇒ ∆ BR(σ) := BR 1 (σ −1 ) × · · · × BR N (σ −N ). 4 It is implicitly assumed that players' mixed strategies are independent; i.e., players do not coordinate.
A strategy σ ∈ ∆ is said to be a Nash equilibrium (NE) if σ ∈ BR i (σ). For convenience, we sometimes refer to a Nash equilibrium simply as an equilibrium.
We say that Γ is a potential game [24] if there exists a function u : Let U : ∆ → R be the multilinear extension of u defined by The function U may be seen as giving the expected value of u under the mixed strategy σ. We refer to U as the potential function and to u as the pure form of the potential function.
Using the definitions of U i and U it is straightforward to verify that Thus, in order to compute the best response set we only require knowledge of the potential function U , not necessarily the individual utility functions (U i ) i=1,...,N . By way of notation, given a pure strategy y i ∈ Y i and a mixed strategy σ −i ∈ ∆ −i , we will write U (y i , σ −i ) to indicate the value of U when player i uses a mixed strategy placing all weight on the y i and the remaining players use the strategy σ −i ∈ ∆ −i .
Given a σ i ∈ ∆ i , let σ k i denote value of the k-th entry in σ i , so that σ i = (σ k i ) Ki k=1 . Since the potential function is linear in each σ i , if we fix any i = 1, . . . , N we may express it as In order to study learning dynamics without being (directly) encumbered by the hyperplane constraint inherent in ∆ i we define where we use the convention that x k i denotes the k-th entry in x i so that . Let X := X 1 × · · · × X N and let T : X → ∆ be the bijection given by T = T 1 × · · · × T N . In an abuse of terminology, we sometimes refer to X as the mixedstrategy space of Γ. When convenient, given an x ∈ X we use the notation x −i to denote the tuple (x j ) j =i . Letting X −i := j =i X j , we define T −i : denote the dimension of X, and note that κ = K, where K, defined earlier, is the cardinality of the joint pure strategy set Y .
Throughout the paper we often find it convenient to work in X rather than ∆. In order to keep the notation as simple as possible we overload the definitions of some symbols when the meaning can be clearly derived from the context. In particular, let BR i : Similarly, given an x ∈ X we abuse notation and write U (x) instead of U (T (x)).
Given a pure strategy y i ∈ Y i , we will write U (y i , x −i ) to indicate the value of U when player i uses a mixed strategy placing all weight on the y i and the remaining players use the strategy x −i ∈ X −i . Similarly, we will say y k i . Applying the definition of T i to (3) we see that U (x) may also be expressed as for any i = 1, . . . , N . We use the following nomenclature to refer to strategies in X.
Definition 2. (i) A strategy x ∈ X is said to be pure if T (x) places all its mass on a single action tuple y ∈ Y . (ii) A strategy x ∈ X is said to be completely mixed if x is in the interior of X.
(iii) In all other cases, a strategy x ∈ X is said to be incompletely mixed.
A fictitious play process is defined as follows.
Definition 3. An absolutely-continuous mapping x : R → X is said to be a fictitious play process with initial condition x 0 ∈ X if x(0) = x 0 and (1) holds for almost all t ∈ R.
Other notation as used throughout the paper is as follows.
• • Suppose m, n, p ∈ N, F i : R m × R n → R, for i = 1, . . . , p. Suppose further that F : R m × R n → R p is given by F (w, z) = (F i (w, z)) i=1,...,p . Then the operator D w gives the Jacobian of F with respect to the components of w = (w k ) k=1,...,m ; that is • A c denotes the complement of a set A, andÅ denotes the interior of A, and cl A denotes the closure of A. • Given an open set Ω ⊂ R n , and k ∈ N ∪ {∞}, C k c (Ω) denotes the set of k-times differentiable functions with compact support in Ω.

Almost All Potential
Games. The set of potential games in which there are N players, each having an action space of size K i , is isomorphic to R Kp , where [28]. We say that a property holds for almost all potential games if the subset of potential games where the property does not hold has L Kp -measure zero.
3. Regular Potential Games. The notion of a regular equilibrium was introduced by Harsanyi [12]. Regular equilibria posses a variety of desirable robustness properties [29].
Being a rather stringent refinement concept, not all games possess regular equilibria. However, "most" games do. A game is said to be regular if all equilibria in the game are regular. Harsanyi [12] showed that almost all N -player games are regular.
The set of potential games forms a low dimensional (Lebesgue-measure-zero) subspace within the space of all games. Since the set of potential games is itself a measure zero set, Harsanyi's regularity result is inconclusive about the prevalence of regular equilibria within this class of games. In a companion paper [28], we study this issue and show the following theorem.
Theorem 4 ( [28], Theorem 1). Almost every potential game is regular. In this paper we will study the behavior of FP in regular potential games. The purpose of this restriction is twofold. First, there are degenerate potential games in which FP does not converge for almost all initial conditions. Restricting attention to regular potential games ensures that the game is not degenerate in this sense. Second, analysis of the behavior of FP is easier near equilibria that are regular. Regularity permits us to characterize the fundamental properties of the potential function U without needing to look at anything higher than second order terms in the Taylor series expansion of U . This substantially simplifies the analysis.
If x * is a regular equilibrium of a potential game, then the derivatives of potential function can be shown to satisfy two non-degeneracy conditions at x * . The first condition deals with the gradient of the potential function at x * and is referred to as the first-order condition; the second condition deals with the Hessian of the potential function at x * and is referred to as the second-order condition. These conditions, introduced in Sections 3.1-3.2 below, will be crucial in the subsequent analysis.
3.1. First-Order Degeneracy. Let Γ be a potential game. Following Harsanyi [12], we will define the carrier set of an element x ∈ X, a natural modification of a support set to the present context. For Let γ i := |C i | and assume that the strategy set Y i , is reordered so that C i = {y 1 i , . . . , y γi i }. Under this ordering, the first γ i − 1 components of any strategy x i with carr i (x i ) = C i are free (not constrained to zero by C i ) and the remaining components of x i are constrained to zero. That is (x k i ) γi−1 k=1 is free under C i and (x k i ) Ki k=γi = 0. The set of strategies {x ∈ X : carr(x) = C} is precisely the interior of the face of X given by (6) Ω Let x * be an equilibrium with carrier C. We say that x * is first-order degenerate if there exists a pair (i, k), i = 1, . . . , N , k = γ i , . . . , K i − 1 such that ∂U (x * ) ∂x k i = 0, and we say x * is first-order non-degenerate otherwise.
Remark 5. We note that using the multi-linearity of U , it is straightforward to verify that an equilibrium is first order non-degenerate if and only if it is quasistrong, as introduced by Harsanyi [12]. In particular, an equilibrium x * is first-order degenerate if and only if carr . . , N . We prefer to use the term first order non-degenerate since it emphasizes that we are concerned with the gradient of the potential function and it keeps nomenclature consistent with the notion of second-order non-degeneracy, introduced next.

Second-Order
Degeneracy. Let C be some carrier set. LetÑ := |{i = 1, . . . , N : γ i ≥ 2}|, and assume that the player set is ordered so that γ i ≥ 2 for i = 1, . . . ,Ñ . Under this ordering, for strategies with carr(x) = C, the firstÑ players use mixed strategies and the remaining players use pure strategies. Assume that N ≥ 1 so that any x with carrier C is a mixed (not pure) strategy.
Let the Hessian of U taken with respect to C be given by Note that this definition of the Hessian restricts attention to the components of x that are free under C We say an equilibrium x * ∈ X is second-order degenerate if the HessianH(x * ) taken with respect to carr(x * ) is singular, and we say x * is second-order non-degenerate otherwise.
Remark 6. Note that both forms of degeneracy are concerned with the interaction of the potential function and the "face" of the strategy space containing the equilibrium x * . If x * touches one or more constraints, then first-order non-degeneracy ensures that the gradient of the potential function is nonzero normal to the face Ω carr(x * ) , defined in (6). Second-order non-degeneracy ensures that, restricting U to the face Ω carr(x * ) , the Hessian of U Ω carr(x * ) is non-singular. If x * is contained within the interior of X, then the first-order condition becomes moot and the second-order condition reduces to the standard definition of a non-degenerate critical point.
Throughout the paper we will study regular potential games. The following lemma from [28] shows that, in any regular potential game, all equilibria are first and second-order non-degenerate. Lemma 12). Let Γ be a potential game. An equilibrium x * is regular if and only if it is both first and second-order non-degenerate.

Potential Production Inequalities.
In this section we prove two key inequalities ( (12) and (13)) that are the backbone of our proof of Theorem 1.
We note that in proving Theorem 1 there is a fundamental dichotomy between studying completely mixed equilibria and incompletely mixed equilibria. Completely mixed equilibria lie in the interior of the strategy space. At these points the gradient of the potential function is zero and the Hessian is non-singular; local analysis of the dynamics is relatively easy. On the other hand, incompletely mixed equilibria necessarily lie on the boundary of X and the potential function may have a nonzero gradient at these points. 5 Analysis of the dynamics around these points is fundamentally more delicate.
In order to handle incompletely mixed equilibria we construct a nonlinear projection whose range is a lower dimensional game in which the image of the equilibrium under consideration is completely mixed. This allows us to handle both types of mixed equilibria in a unified manner.
4.1. Projection to a Lower-Dimensional Game. Let x * be a mixed equilibrium. 6 . . , N } : γ i ≥ 2 , and assume that the player set is ordered so that γ i ≥ 2 for i = 1, . . . ,Ñ . Since x * is assumed to be a mixed-strategy equilibrium, we haveÑ ≥ 1.
Given an x ∈ X, we will frequently use the decomposition ..,Ñ , k=1,...,γi−1 and x p contains the remaining components of x. 7 Let The set of joint pure strategies Y may be expressed as an ordered set Y = {y 1 , . . . , y K } where each element y τ ∈ Y , τ ∈ {1, . . . , K} is an N -tuple of strategies. For each pure strategy y τ ∈ Y , τ = 1, . . . , K, let u τ denote the pure-strategy potential associated with playing y τ ; that is, u τ := u(y τ ), where u is the pure form of the potential function defined in Section 2. A vector of potential coefficients Given a vector of potential coefficients u ∈ R K and a strategy x ∈ X, let 8 Differentiating (5) we see that at the equilibrium x * we have ∂U (x * ) ∂x k i = 0 for i = 1, . . . ,Ñ , k = 1, . . . , γ i − 1 (see Lemma 26 in appendix), or equivalently, By Definition 2, the (mixed) equilibrium x * is completely mixed if γ = κ, and is incompletely mixed otherwise. Suppose γ < κ so that x * is incompletely mixed. Let Since Γ is assumed to be a non-degenerate game, J(x * ) is invertible. By the implicit function theorem, there exists a function g : The graph of g is given by Note that Graph(g) is a smooth manifold with Hausdorff dimension (κ − γ) [9]. An intuitive interpretation of Graph(g) is given in Remark 11. If Γ is a non-degenerate potential game then, using the multilinarity of U , we see that γ ≥ 2 (see Lemma 23 in appendix). This implies that (9) Graph(g) has Hausdorff dimension at most (κ − 2).
Note that P contains the components ofP not constrained to zero. As we will see in the following section, P may be interpreted as a projection into a lower dimensional game in which P(x * ) is a completely mixed equilibrium.

Inequalities.
LetŨ :X → R be given bỹ is the mixed equilibrium fixed in the beginning of the section. LetΓ be a potential game with player set {1, . . . ,Ñ }, mixed-strategy spaceX i , i = 1, . . . ,Ñ , and potential functionŨ . By construction, P(x * ) is a completely mixed equilibrium ofΓ. Moreover, by the definition of a non-degenerate equilibrium, the Hessian ofŨ is invertible at P(x * ).
We are interested in studying the projection P(x(t)) of a FP process into the lower dimensional gameΓ. 9 We wish to show that the following two inequalities hold: for some constant c 1 > 0.
(ii) Suppose (x(t)) t≥0 is a FP process. For x(t) residing in a neighborhood of x * for some constant c 2 > 0. The first inequality follows from Taylor's theorem and the fact that ∇Ũ (P(x * )) = 0. The following section is devoted to proving (13). 10 4.3. Proving the Differential Inequality. We begin with Lemma 8 which shows-roughly speaking-that within the interior of the action space, the FP vector field approximates the gradient field of the potential function.
The following definitions are useful in the lemma.
denote the distance between the set P Xi (B) and the boundary of X i . Since we will eventually be interested in studying a lower-dimensional game derived from Γ, in the lemma we consider an alternative gameΓ of arbitrary size.
where the constant c is given by c = d(P Xi (B), ∂X i ).
Without loss of generality, assume that Y i is ordered so that Differentiating (5) we find that 11 Together with (15), this implies that for k = 1, . . . ,K i − 1 we have Using the multlinearity ofÛ we see that if ξ i ∈ BR i (x −i ) and ξ k i > 0 then y k+1 which is the desired result.
Remark 9. Since the space X i in Lemma 8 is finite dimensional, given any norm · , there exists a constantc > 0 such that For each x = (x p , x m ) ∈ X near to x * , the following lemma allows us to define an additional lower dimensional game Γ xp associated with x p in which the best response set is closely related to the best response set for the original game Γ. The lemma is a straightforward consequence of the definition of the best response correspondence and the continuity of ∇U .
Lemma 10. For x in a neighborhood of x * , the best response set satisfies Given any x = (x p , x m ) ∈ X we defineŨ xp :X → R and Let Γ xp be the potential game with player set {1, . . . ,Ñ }, mixed strategy spacẽ X and potential functionŨ xp . Note that since U is continuous andX is compact, U xp converges uniformly toŨ x * p =:Ũ as x p → x * p . In this sense the game Γ xp can be seen as converging toΓ as x p → x * p . Remark 11. The function g defined in Section 4.1 admits the following interpretation. Suppose we fix some x p = (x k i ) i=1,...,N, k=γi,...,Ki−1 . Then g(x p ) is a completely mixed Nash equilibrium of Γ xp . Moreover, if we let x p → x * p , then the corresponding equilibrium of the reduced game Γ xp converges to x * , i.e., (x p , g(x p )) → (x * p , g(x * p )) = x * , precisely along Graph(g). Remark 12. Suppose x * is a first-order non-degenerate equilibrium. Using the multilinearity of U we see that for any x ∈ X we have carr Due to the ordering we assumed on By Lemma 10, this implies that for all x in a neighborhood of x * and forx The following lemma extends the result of Lemma 8 so it applies in a useful way to the potential functionŨ under the projection P.
Lemma 13. There exists a constant c > 0 such that for all x = (x p , x m ) in a neighborhood of x * and all η ∈ R γi−1 with η sufficiently small we have Proof. By construction, the projection P(x * ) maps x * into the interior ofX. Choose > 0 such that the ball B(P(x * ), ) ⊂X is separated by a positive distance from the boundary ofX. Applying Lemma 8 (and Remark 9) to the game Γ xp we see that there exists a constant c > 0 such that for any x = (x p , x m ) ∈ X with x m ∈ B(P(x * ), ) there holds (22) z Note that the constant in Lemma 8 is only dependent on the distance from the set B (in this case, B(P(x * ), )) to the boundary of the strategy space (in this case,X), and is independent of the particular potential function under consideration-this permits the choice of c > 0 in (22) that holds uniformly for all x p .
By the continuity of U and P we have and P(x) ∈ B(x * , ), for all x in a sufficiently small neighborhood of x * and x ∈ Graph(g). Note that diamX i := max xi, Thus, (22) and (23) give and all x in a neighborhood of x * , x ∈ Graph(g). As long as η ≤ c, the desired result holds for x in a neighborhood of x * , x ∈ Graph(g). But x ∈ Graph(g) =⇒ P(x) = P(x * ) =⇒ ∇ xiŨ (P(x)) = 0, in which case the inequality is trivially satisfied.
Finally, the following lemma shows that the differential inequality (13) holds.
Lemma 14. Let Γ be a non-degenerate potential game with mixed equilibrium x * , and let (x(t)) t≥0 be a FP process. Then the inequality (13) holds for x(t) in a neighborhood of x * . Using the chain rule we may express the time derivative of the potential along the pathP(x(t)) as d dt U (P(x(t))) = ∇U (P(x(t))) I P(x) 0 0 ẋ = ∇ xm U (P(x(t)))(I P(x))ẋ.

Proof. Let
, and let η(t) = (η i (t))Ñ i=1 . Multiplying out the right two terms above we get (24) d dt U (P(x(t))) = ∇ xm U (P(x(t))) (ẋ m + η(t)) By Lemma 10 and Remark 12, if we restrict x(t) to a sufficiently small neighborhood of x * then for any ] −i ) and z i = 0. We note two important consequences of this: (i) If we restrict x(t) to a sufficiently small neighborhood of x * and note that U (P(x(t))) =Ũ (P(x(t)), then by (24) we have (ii) We may force max i=1,...,Ñ η i to be arbitrarily small by restricting x(t) to a neighborhood of x * .
Consequence (i) follows readily by using the definition of FP (1). To show consequence (ii), note that by (1) we haveẋ k i = z k i − x k i for all i = 1, . . . , N , k = 1, . . . , K i , for some z i ∈ BR i (x −i ). But, for x in a neighborhood of x * and k ≥ γ i , we have shown above that z k i = 0, and henceẋ k i = −x k i . 12 Due the ordering we assumed for Y i , we have [x * ] k i = 0 for any (i, k) such that k ≥ γ i . Hence, x k i → 0 as x → x * , for any (i, k) such that k ≥ γ i .
Furthermore, there exists a c > 0 such that | . . , N , ≥ γ j uniformly for x in a neighborhood of x * (see Lemma 27 in appendix). By the definition of η i , this implies that max i=1,...,Ñ η i may be made arbitrarily small by restricting x(t) to a sufficiently small neighborhood of x * . Now, let x(t) be restricted to a sufficiently small neighborhood of x * so that η i (t) is small enough to apply Lemma 13 for each i. Applying Lemma 13 to (25) we get d dtŨ (P(x(t))) ≥ Ñ i=1 c ∇ xiŨ (P(x(t))) for x(t) in a neighborhood of x * . By the equivalence of finite-dimensional norms, there exists a constant c 1 such that d dtŨ (P(x(t))) ≥ c 1 ∇Ũ (P(x(t))) for x(t) in a neighborhood of x * .
Since Γ is assumed to be (second-order) non-degenerate, P(x * ) is a non-degenerate critical point ofŨ . By Lemma 28 (see appendix) there exists a constant c 2 such that c 1 ∇Ũ (x) ≥ c 2 d(x, P(x * )) for allx ∈X in a neighborhood of P(x * ). Since P is continuous we have d dtŨ (P(x(t))) ≥ c 2 d(P(x(t)), P(x * )), for x(t) in a neighborhood of x * .

Proof of Main
Result. We will assume throughout this section that Γ is a regular potential game. By Theorem 4, the ensuing results hold for almost all potential games.
For each mixed equilibrium x * , let the set Λ(x * ) ⊂ X be defined as where g is defined with respect to x * as in Section 4.1.
In this section we will prove Theorem 1 in two steps. First, we will show that for each mixed equilibrium x * , the set Λ(x * ) can only be reached in finite time from an L κ -null set of initial conditions (see Proposition 15), where κ, defined in (4), is the dimension of X. Second, we will show that if a FP process converges to the set Λ(x * ), then it must do so in finite time (see Proposition 22). Since x * ∈ Λ(x * ), Propositions 15 and 22 together show that that for any mixed equilibrium x * , the set of initial conditions from which FP converges to x * has L κ -measure zero.
By Theorem 2 of [28] we see that in regular potential games, the set of NE is finite. Hence, Propositions 15 and 22 imply that FP can only converge to set of mixed strategy equilibria from a L κ -null set of initial conditions. Since a FP process must converge to the set of NE in a potential game ( [4], Theorem 5.5), this implies that Theorem 1 holds.

Finite-Time
Convergence. The goal of this subsection is to prove the following proposition.
Proposition 15. Let Γ be a non-degenerate game and let x * be a mixed-strategy NE of Γ. The set Λ(x * ) can only be reached by a FP process in finite time from a set of initial conditions with L κ -measure zero. That is, Before proving the proposition we present some definitions and preliminary results. Let (26) I . . . , N , k, = 1, . . . , K i , = k, be the set in which player i is indifferent between his k-th and -th actions.
If the game Γ is non-degenerate, then each I i,k, is the union of smooth surfaces with Hausdorff dimension at most (k − 1) (see Lemma 32 in appendix). In particular, for each x ∈ I i,k, there exists a vector ν ∈ R κ that is normal to I i,k, at x. We refer to the set I i,k, as an indifference surface of player i.
We define the setQ ⊆ X as follows. LetQ contain the set of points where two or more indifference surfaces intersect and their normal vectors do not coincide. Furthermore, if an indifference surface I has a componentÎ ⊆ I with Hausdorff dimension less than κ − 1, then we put any points whereÎ intersects with another decision surface intoQ. Since each indifference surface is smooth with dimension at most κ − 1,Q has Hausdorff dimension at most κ − 2. Let As shown in Section 4.1, if x * is non-degenerate, then the set Graph(g) (and hence Λ(x * )) has Hausdorff dimension at most κ − 2. Thus Q has Hausdorff dimension at most κ − 2. 13 The FP vector field (see (1)) is given by FP : X ⇒ X, where Let Z := x ∈ X\Q : x ∈ I i,k, for some i, k, with normal ν at x, (28) and ν · z = 0 for some z ∈ FP(x) .
Since each I i,k, has Hausdorff dimension κ − 1, Z has Hausdorff dimension at most κ − 1. We define the relative boundary of Z, denoted here as ∂Z as follows. If Z has Hausdorff dimension κ−2 or less, then let ∂Z := Z. If Z has Hausdorff dimension κ − 1 then it may be expressed as the union of a finite number of smooth (κ − 1)dimensional surfaces, denoted here as (Z s ) Nz s=1 , 1 ≤ N z < ∞, and a component with Hausdorff dimension at most κ − 2, denoted here as Z . That is, Z = ( Nz s=1 Z s ) ∪ Z . Each Z s , s = 1, . . . , N z is contained in some indifference surface, which we denote here as I s . Define the relative interior of Z s (with respect to I s ) as ri Z s := {x ∈ Z s : ∃ > 0 s.t. B(x, )∩I s ⊂ Z s }, and define the relative boundary of Z s as ∂Z s := cl Z s \ri Z s . We then define the relative boundary of Z as ∂Z := Nz s=1 ∂Z s ∪ Z . 13 Proposition 15 can easily be generalized to say that any set A ⊂ X such that cl A has Hausdorff dimension at most κ − 2, can only be reached in finite time from a set of L κ -measure zero by substituting A for Λ(x * ) throughout the section.
Note that ∂Z is a set with Hausdorff dimension at most κ − 2. By Lemma 35 in the appendix, the FP vector field is oriented tangentially along Z, in the sense that for any x ∈ Z there holds ν · y = 0 for any vector ν normal to Z at x, and any y ∈ FP(x). This implies that FP paths can only enter or exit Z through ∂Z. Let The following technical lemma will be used to show that FP is well posed within X * (see Lemma 17). It is a consequence of the fact that the FP vector field can only have jumps that are tangential to indifference surfaces.
Lemma 16. Suppose x ∈ X * is in some indifference surface I i,k, . Then there exists a constant c > 0 and a vector ν that is normal to I i,k, at x, such that for allx ∈ X * in a neighborhood of x.
Proof. By the definition of I i,k, , if x ∈ I i,k, then for allx ∈ X such that x −i = x −i we havex ∈ I i,k, . This implies that for any vector ν that is normal to I i,k, , the (i, m)-th component of ν must be zero for all m = 1, . . . , K i − 1.
Suppose that x ∈ X * ∩ I i,k, . Since x ∈ Q, there is a neighborhood of x in which no indifference surface intersects with I i,k, . This implies that forx within a neighborhood of x, Together, these two facts imply that for allx in a neighborhood of x, we have ν · z = ν · z for all z ∈ BR(x), z ∈ BR(x), for any vector ν that is normal to I i,k, at x. Since x / ∈ Z, recalling the form of FP (27), this means we can choose a vector ν that is normal to I i,k, at x and a constant c > 0 such that ν · z > c for z ∈ FP(x) for allx in a neighborhood of x.
The following lemma gives a well-posedness result for FP inside X * .
Proof. If x ∈ X * is not on any indifference surface, then FP is single valued in a neighborhood of x, and (1) is (locally) a Lipschitz differential equation with unique local solution.
Suppose that x 0 ∈ X * is on an indifference surface I. By Lemma 16 there exists a constant c > 0 such that for allx in a neighborhood of x we have FP(x) · ν > c, where ν is a normal vector to I at x. This implies that for δ > 0 sufficiently small we have {t ∈ [−δ, δ] : x(t) ∈ I} = {0}. Furthermore, since x ∈ Q, for δ > 0 sufficiently small we have (29) {t ∈ [−δ, δ] : x(t) ∈ I i,k, , for any i, k, } = {0}. Now, let x 0 ∈ X * and let (x(t)) t≥0 and (z(t)) t≥0 be two solutions to (1) with x(0) = z(0) = x 0 . If (x(t)) t≥0 never crosses an indifference surface, then the flow is always classical and the two solutions always coincide; i.e., x(t) = z(t), t ≥ 0. Suppose that (x(t)) t≥0 does cross an indifference surface and let t * ≥ 0 be first time when such a crossing occurs. For t < t * , the flow is classical and we have x(t) = z(t) for t ≤ t * .
Sincex =ẑ, and since the time-reversed flow is classical for 0 ≤ τ < δ (in particular, of the formẋ = a + x for some constant a), we getx(δ) =z(δ). But this is impossible because the paths (x(t)) t≥0 and (z(t)) t≥0 are absolutely continuous and we already established thatx(δ) = x(t * ) = z(t * ) =z(δ). As a matter of notation, we say that λ is a signed measure on R κ if there exists a Radon measure µ on R κ and a µ-measurable function σ : for all compact sets K ⊂ R κ . When convenient, we write σµ to denote the signed measure λ in (30).
Letting elements x ∈ X be written componentwise as (x s ) κ s=1 , we recall [9] that a function u ∈ L 1 (Ω) (with Ω ⊆ R κ , Ω open) is a function of bounded variation (i.e., a BV function) if there exist finite signed Radon measures D s u such that the integration by parts formula The measure D s u is called the weak, or distributional, partial derivative of u with respect to x s . We let Du := (D s u) s=1,...,κ .
Here Ju is supported on a set J u with Hausdorff dimension κ − 1, and Cu is singular with respect to L κ and satisfies Cu(E) = 0 for all sets E with finite H κ−1 measure. The L 1 function ∇u is analogous to a classical derivative, and in particular if u is differentiable on an open set V then Du = ∇uL κ on that set, with ∇u matching the classical derivative. Furthermore, if u jumps across a smooth (κ − 1)-dimensional hypersurface, then for x on the hypersurface we have where u + is the value of u on one side of the surface, u − is the value on the other, and ν is the normal vector pointing from u − to u + [1]. A vector-valued function f ∈ L 1 (Ω : R κ ) is a function of bounded variation if each of its components is also of bounded variation. Letting f be written componentwise as f = (f s ) κ s=1 , we write Df := (D j f s ) j,s=i,...,κ . Next we define the divergence of a function f ∈ L 1 (Ω : R κ ), denoted by D · f , as the measure Given a constant c ∈ R, we say that D · f = c if D · f = dD·f dL κ L κ , and dD·f dL κ = c, where dD·f dL κ denotes the Radon-Nikodym derivative. The following lemma characterizes the divergence of the FP vector field. As a matter of notation, if a function f : X → X satisfies f (x) ∈ FP(x) for all x ∈ X then we say f ∈ FP.
Lemma 18. For every f ∈ FP, the vector field f satisfies D · f = −1.
The proof of this lemma follows from the fact that FP is piecewise linear, and any jumps in FP are tangential to indifference surfaces.
Proof. Suppose f ∈ FP, and let f be written componentwise as Since f k i is piecewise linear and only jumps on the set Ki =1, =k I i,k, which has finite κ−1 measure, f k i has no Cantor part; that is, Cf k i = (C j, f k i ) j=1,...,N, =1,...,γj = 0 (see (32)). Hence, the singular component of D · f , which we denote here as S, has no Cantor part and is given by Suppose that x ∈ I i,k, for some (recall = k). Suppose ν is a vector that is normal to I i,k, at x. By the definition of I i,k, , if x ∈ I i,k, then for allx ∈ X such thatx −i = x −i we havex ∈ I i,k, . This implies that the (i, k)-th component of ν must be zero. Since Since this is true for every pair (i, k) we see that S = 0, and hence D · f = −1 in the interior of X. An identical argument holds on the boundary of X, and hence, S = 0 and D · f = −1.
The following lemma shows that for sets E ⊆ X * with relatively smooth boundary, the surface integral of FP over the boundary of E is well defined.
Lemma 19. Let E be a subset of X * with piecewise smooth boundary. For any functions f, g ∈ FP we have where ν E denotes the outer normal vector of E.
Proof. Suppose x ∈ X * is not on any indifference surface I i,k, . Then FP(x) maps to a singleton and f (x) = g(x).
Suppose x ∈ X * is on an indifference surface I i,k, . Let ν I denote a normal vector to I i,k, . Since x ∈ X * , the vector field FP can only jump tangentially to ν I . Using similar reasoning to the proof of Lemma 16, this implies that for any a, b ∈ FP(x) we have a · ν I (x) = b · ν I (x). Hence FP(x) · ν := a · ν, a ∈ FP(x) is well defined for such x.
In particular, note that if x ∈ X * is on some indifference surface I and ν I = ν E at x, then f (x) · ν E = FP(x) · ν I for any function f ∈ FP.
Let I be the union of all indifference surfaces. Since ∂E is piecewise continuous and the indifference surfaces are smooth, the set S := {x ∈ X * : x ∈ I ∩ ∂E, ν I (x) = ν ∂E (x)} has H κ−1 -measure zero, where ν I (x) and ν ∂E (x) denote the normal vectors to I and ∂E at x.
We have shown that f (∂E)\S = g (∂E)\S for all f, g ∈ FP, and H κ−1 (S) = 0, and hence, for all f, g ∈ FP.
The following lemma shows that, within X * , the FP vector field compresses mass at a rate of −1. In particular, this implies that, within X * , FP cannot map a set of positive measure to a set of zero measure in finite time. 14 Lemma 20. Let E be a compact subset of X * with piecewise smooth boundary and finite perimeter. Then where ν E denotes the outer normal vector of E.
Proof. We first note that by Lemma 18 for every f ∈ FP we have E dD · f = −L κ (E).
Let (f n ) n≥1 , f n : X * → X * be a sequence of uniformly bounded C 1 functions such that f n → f a.e. for some function f : X * → X * satisfying f (x) ∈ FP(x) for all x ∈ X * . (Such a sequence can be explicitly constructed by smoothing the FP vector field, e.g., [10].) Let f and each f n be written componentwise as f = (f s ) κ s=1 and f n = (f s n ) κ s=1 . Let D · f n = κ s=1 D s f s n be the divergence measure associated with f n and D · f = κ s=1 D s f s the divergence measure associated with f . Since f and f n are BV functions, by (31) we have Hence, where the third line follows from the Gauss-Green theorem [9], the fourth line follows from the dominated convergence theorem (by assumption, E has finite perimeter and a piecewise smooth boundary, and f is bounded), and the fifth line follows from Lemma 19.
We now prove Proposition 15.
Proof. We begin by noting that, by Lemma 34 in the appendix, cl Q, has Hausdorff dimension at most κ − 2.
Let > 0. By the definition of the Hausdorff measure ( [9], Chapter 2), there exists a countable collection of balls (B j ) j≥1 , each with diameter less than , such Letting R denote the flux through ∂B into E (T − t) and again letting ν denote the outer normal to ∂E (T − t), for t > 0 we have where the third line follows by Lemma 20. Noting that FP ∞ < ∞, the flux through ∂B is bounded by By (36) we have H κ−1 (∂B ) → 0 as → 0, and henceR → 0 as → 0.
Using the integral form of Gronwall's inequality, (37) and (38) give V (t) ≤ tR e t , 0 < t ≤ T . In particular, this means that where the right hand side goes to zero as → 0. Sending → 0, we see that the set Since paths may only enter Z through the boundary ∂Z, this means that the set of points in X from which Z can be reached within time T is contained in W (T ) ∪ Z. Furthermore, the set of points from which Q ∪ Z can be reached within time T is contained in W (T ) ∪ Z ∪ Q, which is a L κ -measure zero set. Since this is true for every T > 0, we get the desired result.
Remark 21. We note that the proof given above implies that Q ∪ Z can only be reached in finite time from a L κ -measure zero subset of X * . Since X * := X\(Q ∪ Z), this, combined with Lemma 17, implies that for almost every initial condition x 0 ∈ X * , there exists a unique FP process x with x(0) = x 0 . Since L κ (X\X * ) = 0, this implies that for almost every x 0 ∈ X, there exists a unique FP process x with x(0) = x 0 .

Infinite-Time
Convergence. The following proposition shows that it is not possible to converge to Λ(x * ) in infinite time.
Proposition 22. Let Γ be a non-degenerate potential game and let x * be a mixedstrategy equilibrium. Suppose (x(t)) t≥0 is a FP process and x(t) → x * . Then x(t) converges to Λ(x * ) in finite time.
Using ( Let t be fixed. Recursively applying the above inequality we find that Thus if P(x(t)) converges to P(x * ), it must reach it for the first time in finite time.
By construction P(x) = P(x * ) if and only if x ∈ Λ(x * ). Hence, if x(t) converges to Λ(x * ) it must reach it for the first time in finite time.

Appendix.
Lemma 23. Suppose Γ is a degenerate game. At any mixed equilibrium there are at least two players using mixed strategies.
Proof. Suppose that x * is an equilibrium in which only one player uses a mixed strategy-say, player 1. Let C i = carr i (x * ) and γ i = |C i |. Then the mixed strategy Hessian is given by ..,γi = 0, where the equality to zero follows since U is linear in x 1 .
Proof. The result follows readily from (43).
Lemma 26. Suppose x * is an equilibrium and y k i ∈ carr(x * ), k ≥ 2. Then Proof. Since U is multilinear, y k i must be a pure-strategy best response to x * −i . The result then follows from Lemma 24.
Proof. We will prove the result using an inductive argument. Suppose first that n = 1 so that p : R → R. Let k denote the degree of p. Since p is not identically zero, the fundamental theorem of algebra implies that p has at most k zeros. Hence L 1 (Z(p)) = 0. Now, suppose that n ≥ 2 and for any polynomialp : R n−1 → R there holds L n−1 (Z(p)) = 0. We may write where k is the degree of p in the variable x n , x = (x 1 , . . . , x n−1 ), the functions p j , j = 0, . . . , k are polynomials in n − 1 variables, and where at least one p j is not identically zero.
If (x, x n ) is such that p(x, x n ) = 0 then there are two possibilities: Either (i) p 0 (x) = . . . = p k (x) = 0, or (ii) x n is the root of the one-variable polynomial p x (t) := k j=1 p j (x)t j .
Let A and B be the subsets of R n where (i) and (ii) hold respectively, so that Z(p) = A ∪ B. For any x n ∈ R we have (x, x n ) ∈ A ⇐⇒ x ∈ Z(p j ), ∀j = 1, . . . , k. By the induction hypothesis, we have Z(p j ) = 0 for at least one j, and hence R n−1 χ A (x, x n ) dx = 0 for any x n ∈ R, where we include the argument in the characteristic function χ A , in order to emphasize the dependence on both x and x n . This implies that x n → R n−1 χ (x,xn)∈A dx is a measurable function (it's identically zero) and By the fundamental theorem of algebra, for any x ∈ R n−1 there are at most k values t ∈ R such that (x, t) ∈ B, and hence R χ B (x, x n ) dx n = 0. As before, this implies that x → R χ B (x, x n ) dx n is a measurable function and Since Z(p) = B ∪ A, this proves the desired result.
Remark 31. Note that if p ≡ 0, then Z(p) = R n . Thus, in general, if p : R n → R is a polynomial, then Lemma 30 implies that either Z(p) = R n or L n (Z(p)) = 0.
Lemma 32. Suppose Γ is a non-degenerate potential game. Then each indifference surface I i,k, , as defined in (26), is a union of smooth surfaces with Hausdorff dimension at most κ − 1.
Proof. Throughout the proof, when we refer to the dimension of a set we mean the Hausdorff dimension. Let i ∈ {1, . . . , N }, k, ∈ {1, . . . , K i }, k = and let I := I i,k, , where I i,k, is as defined in (26). Note that I is the zero-level set of the polynomial p(x) := U (y k i , x −i ) − U (y i , x −i ). By Lemma 30 and Remark 31 we see that either L κ (I) = 0, or I = X. Being the level set of a polynomial, if L κ (I) = 0, then I is the union of smooth surfaces with dimension at most κ − 1.
Suppose that I has dimension greater than κ − 1. Then by the above, we see that I = X. Since Γ is a finite normal-form game, there exists at least one equilibrium x * ∈ X. Letting x * be written componentwise as x * = ([x * ] m j ) j=1,...,N, m=1,...,Ki−1 we see that if [x * ] k i > 0, then x * ∈ I = X implies that x * is a second-order degenerate equilibrium. Otherwise, if [x * ] k i = 0, then x * ∈ I = X implies that x * is a first-order degenerate equilibrium. In either case we see that x * is a degenerate equilibrium, and hence Γ is a degenerate game, which is a contradiction.
Since I was an arbitrary indifference surface, we see that if Γ is a non-degenerate game, then every indifference surface has dimension at most κ − 1.
Lemma 34. Let Q be defined as in Section 5.1. Then cl Q has Hausdorff dimension at most κ − 2.
Proof. Let A be the subset of X where two or more decision surfaces intersect. Let N ⊂ A be the subset of X where two or more decision surfaces intersect and their normal vectors coincide. Define the relative interior of N with respect to A as ri N := {x ∈ N : ∃ > 0 s.t. B(x, ) ∩ A ⊂ N }, and define the relative boundary of N with respect to A as ∂N := cl N \ri N.
Since each indifference surface has Hausdorff dimension κ − 1, N has Hausdorff dimension at most κ − 1. In particular, N is the union of a finite number of smooth κ − 1 dimensional surfaces and a component with Hausdorff dimension at most κ − 2. This implies that the relative boundary of N has Hausdorff dimension at most κ − 2.
LetQ be as defined in Section 5.1. Note that the closure ofQ satisfies clQ ⊆ Q ∪ ∂N . Since the setsQ and ∂N have Hausdorff dimension at most κ − 2, the set clQ also has Hausdorff dimension at most κ − 2.
Let Λ(x * ) be as defined in Section 5. If Λ(x * ) = {x * }, then Λ(x * ) is closed and has Hausdorff dimension 0. Otherwise, Λ(x * ) is defined as the graph of g. In Section 4.1 it was shown that Graph(g) has Hausdorff dimension at most κ − 2. Since g is a smooth function, the closure of Graph(g) has Hausdorff dimension at most κ − 2.
Recall that Q is defined as Q =Q ∪ Λ(x * ) and hence cl Q = clQ ∪ cl Λ(x * ). Since clQ and cl Λ(x * ) each have Hausdorff dimension at most κ−2, cl Q also has Hausdorff dimension at most κ − 2.
Lemma 35. Let Z be as defined (28). Then for any x ∈ Z there holds ν · y = 0 for any vector ν normal to Z at x, and any y ∈ F P (x) Proof. Suppose x ∈ X and x is in some indifference surface I i,k, . Suppose ν is a vector that is normal to I i,k, at x. By the definition of I i,k, , if x ∈ I i,k, then for allx ∈ X such thatx −i = x −i we havex ∈ I i,k, . This implies that the (i,k)-th component of ν must be zero for everyk = 1, . . . , K i − 1.
For x ∈ X, let N (x) := {(i, k) : i ∈ {1, . . . , N }, k ∈ {1, . . . , K i − 1}, x ∈ I i,k, for some = 1, . . . , K i − 1, = k}. Letting FP k i be the (i, k)-th component map of FP, note that by the definition of an indifference surface, FP k i (x) is single valued for every pair (i, k) / ∈ N (x). Suppose x ∈ X\Q is in at least one decision surface I and let ν be a vector that is normal to I at x. Note that x / ∈ Q implies that if x is contained in any other decision surfaceÎ = I, then ν is also normal toÎ at x. Letting ν be written componentwise as ν = (ν k i ) i=1,...N, k=1,...,Ki−1 , the above discussion implies that ν k i = 0 for every pair (i, k) ∈ N (x).
Now suppose x ∈ Z. By the definition of Z we have x / ∈ Q and x is in at least one decision surface I. Let ν be a vector that is normal to I at x. By the definition of Z, there exists some y ∈ FP(x) such that y · ν = 0. Breaking this down in terms of components in N (x) we have 0 = y · ν = (i,k)∈N (x) The first sum is zero since ν k i = 0 for all (i, k) ∈ N (x). Consequently, the second sum must also be zero. But we have shown above that F k i (x) is single valued for any (i, k) / ∈ N (x). Hence, for anyỹ ∈ FP(x) we haveỹ k i = y k i for all (i, k) / ∈ N (x), and in particular, Sinceỹ ∈ FP(x) was arbitrary, this proves the desired result.


Personal Tax Relief Schemes: Awareness and Usage among Personal Income Taxpayers within Kumasi Metropolis of Ghana

To lessen burdens caused by enforcement of tax laws and to ensure equal distribution of tax burdens, good tax systems grant individual personal income taxpayers (PIT) reduction(s) in their taxable income through personal income tax reliefs schemes (PTRS). The Ghanaian tax system is no exception to this. The usage however depends on the taxpayer’s awareness and willingness to adopt. This study therefore sought to ascertain the PTRS awareness and usage level and the factors that influence the PTRS usage in mitigating tax burdens among individual Ghanaian PIT within the Kumasi Metropolis. Also, this study sought to impale research interest in PTRS usage considering its effects on government tax revenue and disposable household and national incomes. The study used a survey research design with sets of structured questionnaires. The results of the study are based on responses from 210 PIT from Kumasi Metropolis of Ghana analyzed using descriptive statistics and binary logistic regression. The results suggest that while there is fair level of PTRS awareness, PTRS usage remains very low among Ghanaian PIT from Kumasi Metropolis. The study also identified knowledge, awareness, challenges, and perception as statistically key factors that influence PTRS usage in mitigating tax burdens among individual PIT from Kumasi Metropolis. Consequently, this paper recommends that Ghana Revenue Authority and/or tax policy makers, social advocators and National Commission on Civic Education among others pay attention to these factors so as to step up interest in PTRS usage as means of mitigating tax burdens if voluntary tax compliance is to be encouraged.

INTRODUCTION
Review of economics literature suggests that guaranteeing adequate disposable income in the hands of individual households and at the same Original Research Article time, ensuring equitable distribution of disposable income in every economy have for many years remained one of the most significant themes in public finance management. In fact, the issues of adequate disposable income in the hands of households and its equitable distribution are still dominant issue in fiscal policies even in today's economic environment as nations strive to maximize tax revenue as against other forms of sources of government revenue [1]. According to literature, this stems back from the days of Adam (1776) and has remained one of the most important economic and fiscal policies of governments. This has implications for not only government revenue, but also trade (demand and supply), savings and investment which also have resultant effect on growth and national income [1]. However, tax and/or taxation may be seen as the most basic determinant of equitable distribution of wealth or resources among households in any jurisdiction ( [2], [3], [4], [5]).
Notably, enforcement of taxes by tax officials causes individual taxpayers to give up part of their control over private resources (that is income) to the government of a jurisdiction ( [1], [2], [3], [4], [5], [6]). This given up resource, known as tax revenue, is one of the most crucial sources of a government's revenue for the production of common goods and services for the general citizenry ( [1], [2], [3], [4], [5], [6], [7], [8]). In giving up those controls, there is huge pinch usually suffered by those who paid the taxes known as tax burden. To offload the said burden caused by enforcement of the tax law, many taxpayers resort to income tax evasion ( [1], [2], [3], [4], [5], [6]). Evasion, however, has negative effects on both the taxpayer(s) and the tax revenue accruing to the government ( [1], [2], [3], [4], [5], [6]). As it were, as a taxpayer engages in tax evasion, he may be regarded as disloyal citizen as well as infringing the tax status. These may be punishable by imprisonment, fine or both. The net likely effect will be that the taxpayer will be more disadvantaged when caught in the act of tax evasion. On the other hand, the tax revenue due the government will be negatively affected as result of non-payment of the imposed taxes by the taxpayers due to evasion. Consequent to that, as the taxes are not being paid, the little tax revenue collected as well as other resources will be spent combating the crime of evasion.
Consequently, in order to lessen the burden of taxes on individuals, avoid the occurrence of evasion, ensure equal distribution of the tax burden among various taxpayers, and also to elicit voluntary tax compliance, good tax systems across the globe usually grant taxpayers a reduction in their taxable base ( [2], [4], [5], [6]). This reduction in the taxable base is technically referred to as personal tax reliefs (PTR) and this is built into Ghanaian tax systems. According to literature, opinions are that most individual taxpayers as rational would want to take full advantage of tax incentives including PTR in reducing their tax burden as provided for in the tax laws ( [2], [4], [5], [6], [9], [10]). However, the reverse is the case among Ghanaian individual income tax payers ( [6], [11]).
The knowledge that tax officials enforce the tax law in exaction and nothing more than what the tax law requires helps taxpayers to make the most of the provisions of the tax law to their advantage in order to reduce one's tax liabilities ( [2], [4], [5]). This has assumed some popularity not only among taxpayers, but also the judiciaries as postulated by Judge Learned Hands in [12] and Lord Tomlin in [13].

The Research Aim and Objectives
Considering importance of taxation and the effects of non-compliance on government cash flow, there have been enormous tax compliance researches across the globe ( [1], [3]). However, careful review of extant literature suggests PTRS research in developing economies specifically in Ghana is still at its infancy resulting in paucity in PTRS literature. Also, until the assertion by the Minister of Finance and Economic Planning in the 2012 Budget Statement presentation, it appears there has not been any PTRS research in Ghana despite PTRS importance to both government's tax revenue and households' disposable income and/or level of investment. In Ghanaian context, the only empirical PTRS research available (as at March, 2015) was [6] which was to corroborate the Minister's assertion. This study thus concluded that, although most Ghanaian employees are sentient of the PTR schemes, only few of them have ever taken advantage of PTRS as means of reducing their tax burden so far.
A careful review of [6] study revealed a conspicuous omission of factors that could influence taxpayers' likelihood of PTRS usage in mitigating tax burdens but rather the study was quick to recommend further studies to investigate these. Further, [6] study only concentrated on employee taxpayers, leaving out the selfemployed taxpayers. It is therefore imperative to postulate a study that probes awareness and usage of personal tax relief scheme(s) among both the self-employed and employee taxpayers in Ghana. The aim of this study, however, is to take a slight paradigm shift in its empirical investigation of the level of awareness of personal tax relief scheme using a sample comprehensive enough to include both selfemployed and employee taxpayers. This study therefore sought to ascertain and/or predict the PTRS awareness and usage level as well as the factors that influence PTRS usage in mitigating tax burdens among individual Ghanaian personal income taxpayers within the Kumasi Metropolis. It is anticipated that this study will fill the existing research gap and thereafter impale research interest in the issue under study while answering [6] call.

Research Questions
Based on the foregoing objectives, this study sought to find answers to the following main research questions: 1. What is the level of awareness and usage of PTRS in lessening tax burdens among Ghanaian taxpayers? 2. Are there any significant relationships between Ghanaian taxpayers level of PTRS usage and their demographic factors?

REVIEW OF LITERATURE
PTR may be defined as the statutory deductions from assessable income in ascertaining the chargeable income of an individual ( [2], [4], [5], [6]

Model Specification
Extant taxation literature consistent with utility theory asserts that compliance with tax laws or otherwise is likely to be influenced by some underlying factors ( [8], [15], [16]). In spite of the increasing importance of the above mentioned assertion, such factors usually differ depending on the individual taxpayer's circumstance. Also, various tax compliance studies concluded that certain factors best influence behaviour pattern (specifically tax compliance) ( [1], [8], [15], [16], [17], [18], [19]). As usage of PTRS is provided for in the tax laws, we could conjecture that, PTRS usage in mitigating tax burdens within the confines of the tax law (instead of unapproved evasion) may be seen as tax compliance. Accordingly, taxpayers' PTRS usage in mitigating tax burdens can be explained by personal or demographic, institutional, and other environmental factors such as gender; age; educational background; nature and/or type of work; experience; similarities in taxpayer's prior engagements to current one; knowledge of the tax law provisions; experience in payment of tax(es); awareness and/or consciousness; perceptions of tax law provisions; challenges; among others. However, these drivers may not be constant and are entirely shrouded.
For instance, the gender (Genditp) of the respondents is included because though both sexes are expected to pay tax on income earned from their engagements, more males are engaged in employment than their female counterparts. Impliedly, more males are more likely to use PTRS in mitigating their tax liabilities than their female counterparts.
However, while some studies suggest that female taxpayers conformed and complied with tax rules more than male taxpayers [17], others like [20] argued that the differences in the compliance behaviour between males and females may be narrowed as more nontraditional generation of women is evolving in our time.
A dummy variable was used to specify the gender of the respondents. A value of 1 was assigned to males and 2 to females. It is expected that when an income taxpayer is a male, he will be more likely to use PTRS than if he is a female income taxpayer. Age (Ageitp) is specified as individual taxpayer's age measured in years at the time of data collection (March, 2015).
Educational background enables individual income taxpayers (EducBitp) to comprehend more easily complex legislative and general information, keep records, conduct basic analysis and interpretation so as to make the right decisions as to comply or otherwise. Literature seems to portray that educational background has a significant impact on attitude to tax compliance behaviour of taxpayers ( [1], [21], [22], [23], [24]). Thus, well educated individual income taxpayers are assumed to be more knowledgeable in most legislations and hence aware of PTRS usage. Educational backgrounds of respondents were specified as applicable, not applicable. It is hypothesized that individual income taxpayers with adequate educational background (or with higher levels of education) are more likely to be aware of PTRS as under the tax laws of Ghana and hence have high tendency to avail to PTRS.
The working experience of individual taxpayer (Ageweitp) measured in terms of number of years is believed to positively influence PTRS usage of individual income taxpayers. Thus, those with many years working experience may have been aware of PTRS, might have ever utilized it and are more likely to subsequently utilize it in mitigating their tax liabilities.
Nature and/or type of work of an individual income taxpayer may have different implications for PTRS usage. As employee income taxpayers are granted upfront reliefs, self-employed taxpayers are granted reliefs at the end of their basis period. This may have consequent influence on income taxpayers' PTRS usage. Nature and/or type of work of an individual taxpayer (Natworkitp) is specified as dummy variable and a value of 1 is assigned to working and 2 for not working. It is expected that selfemployed taxpayers with large taxable base outside the minimum threshold are less likely to utilize PTRS than employee taxpayer [1].
According to literature, prior similar experience has great tendency to influence behavior patterns [25]. Drawing on this assertion, this study posits that an employee's previous experience in PTRS usage is likely to influence his future usage and hence its inclusion in the model. Similarities of prior engagements to current one (SimPriorEng) specified as dummy variable and a value of 1 is assigned to similar prior engagement or working and 2 as not similar prior engagement may accord the taxpayer the opportunity to follow previous tax history. In this case making it easier for the taxpayer to rely on earlier information and practice in tax obligations. Thus, taxpayers who ever got assistance from her/is previous employer in utilizing PTRS is more likely to do same in similar engagement.
Also, prior experiences in paying taxes by the individual taxpayer (Pexpaytax) specified as dummy variable and a value of 1 is assigned to heavy tax burden and 2 as otherwise; may have significant positive effect on taxpayers. Thus, taxpayers with heavy tax burdens are more likely to deploy PTRS than those with otherwise less tax burdens.
Taxpayers' knowledge in tax laws is said to have significant relationship with tax compliance [16]. Some income taxpayers (such as accountants, lawyers, auditors, tax officers, etc.) are regarded as knowledgeable in the income tax law (Knoitp). It is therefore assumed that when an income taxpayer exhibits adequate knowledge in the income tax law, s/he is more likely to use PTRS in reducing his/her taxable base and hence pay less tax. This variable is assigned 1 if the taxpayer has adequate knowledge and 2 otherwise. It is expected to have positive relationship with probability of PTRS usage.
There is highly likelihood of PTRS usage among individual taxpayers who are very much aware of the capability of PTRS (Awareness) in helping them largely reduce their tax burdens. In other words, income taxpayers who are aware of PTRS are more likely to deploy PTRS than those who are otherwise unaware. This variable is assigned 1 if the taxpayer is aware of PTRS as under the tax law and 2 otherwise not aware. It is expected that awareness should have positive relationship with probability of taxpayer's PTRS usage.
Similarly, challenges encountered by individual taxpayers in their prior attempt(s) if any (challenges) when accessing reliefs is likely to influence his subsequent PTRS usage in reducing tax burden. The challenges therefore specified as dummy variable with a value of 1 if regarded as cumbersome and 2 if otherwise.
Prior personal experiences and/or challenges encountered by individual income taxpayers are likely to cause the taxpayer to form a personal perception or opinion on a tax system. Thus, perception(s) and/or opinion form on tax system based on experiences or challenges are expected to have great influence on one's usage or patronage of the said tax system.
As it were, any individual income taxpayer who wishes to mitigate his/her tax burdens legally within the confines of the tax law, and more importantly without having any serious issues with the law, will ultimately utilize the PTRS instead of tax evasion. Therefore, the dependent variable denoted as the PTRS usage in mitigating tax burdens by individual income taxpayers takes the value of 1 if the individual income taxpayer uses PTRS and 2 if otherwise.
Stemming from the above discussions, this study adopts the binary logistic regression modeling technique as the analytical framework for predicting the Odds of PTRS usage decisions among the study respondents. According to literature, there are numerous models for estimating determinants in practice such as probit (standard normal), logit (logistic), tobit (extreme value) ( [8], [17], [19], [26]) and structural equation model [18]. The choice of this methodology for this study, however, is as a result of the fact that the regressors of the model are binary and that it has great potency to predict categorical outcome using multiple categorical and/or numeric predictors.
Guided by related prior studies, the logistic regression model is thus specified as: = is the probability that a respondent will use PTRS in mitigating his(er) tax burden or not; μ = Error or random disturbance term; ƛ 0 = Constant term; ƛ 1 ... ƛ 10 = The logistic regression coefficients estimated.
However, these coefficients do not provide direct information about the effect of the changes in the independent variables on the PTRS, but on the probability of respondents applying the PTRS as provided under the tax laws. The regression analysis was carried out between the dependent variable (PTRS usage in mitigating tax burden) and the independent variables as specified in Table 2. Based on prior studies social economic variables such as gender of the individual taxpayer; age; educational background; number of years working experience; nature and/or type of work; similarities of prior and current engagements; prior experiences in paying taxes; knowledge of the tax laws; perception of the procedures for applying for PTRS; and challenges encountered in the previous attempts to apply PTRS were used as the independent variables (as specified in Table 2) as they are believed to influence behaviour ( [1], [3], [18]).
As indicated above, any individual income taxpayer who wishes to mitigate his/her tax burdens legally within the confines of the tax law, and more importantly without having any serious issues with the law, will ultimately utilize the PTRS instead of tax evasion. Therefore, the dependent variable denoted as the PTRS usage in lessening tax burdens by individual income taxpayers takes the value of 1 if the individual income taxpayer uses PTRS and 2 if otherwise.
As defined above, P i is the probability of a taxpayer choosing to utilize PTRS to mitigate his/her tax burden (that is the dependent variable) and In is a linear combination of the natural logarithm of the independent variables ( [18], [27]). As coefficient results are not quite easily interpretable, the natural logarithm of the coefficient (ExpB, that is the odd ratio) could be taken as the times the likelihood increases or decreases with every change in value of one of the value of the independent variables [18]. Thus, should an independent variable have an ExpB or odd ratio of 5, then the likelihood of utilizing PTRS will increase by fivefold for every one increase in the value of taxpayer's PTRS usage for the independent variable. On the other hand, if an independent variable has an ExpB or odd ratio of 0.20, the likelihood of Taxpayer's PTRS usage is only one-fifth as large as a taxpayer who enters a value one point greater in the independent variable.

METHODOLOGY
To obtain the data for analysis, this study used a survey research approach where sets of selfadministered questionnaires were used as data collection instruments. The population for the study was basically every Ghanaian within the Kumasi Metropolis who is qualified to pay tax under the income tax laws of Ghana specifically [27] as amended. In a similar study on the current issue by [6], a sample of 219 respondents from Greater Accra, Ashanti and Brong Ahafo regions was used. For the purposes of this study, a sample of respondents from Kumasi, the capital of Ashanti Region, was used based on the estimation method as given by [28] and [29] to determine the sample size for the study as shown below: According to [6] study, about 19.20% of the 78.1% taxpayers who are aware of the existence of the personal tax relief scheme under the tax laws of Ghana, have taken advantage of the scheme. In other words, about twenty-five percent of those who were aware of the existence of the scheme in [6] study used personal tax relief scheme(s) as a means of reducing their tax liabilities (i.e. burdens). Based on the above assertions, we thus assumed 95% confidence level and 5% margin of error to estimate our sample for the study as follows:  [30] as cited in ( [29], [31], [32). Therefore, the sample size of 236 used is reasonable and representative for this study. The data were obtained primarily from the field survey conducted within Kumasi and its suburbs, namely, Abuakwa, Tanoso, Kwadaso, Danyame, Adum, Harper Road, Ridge, Ahodwo Roundabout using a structured questionnaire. The data collected were analyzed using SPSS Version 21 with help from MS Excel Application.
However, at the field due to the vastness of the population, a purposive and convenient sampling approach was used. The sample comprised members of universities' communities (both private and public); employees of financial institutions; local government staff; employees in the private sector; and self employed individuals. These respondents were selected for this study as they are perceived as relatively regular taxpayers ([6], [33]) and hence maybe knowledgeable and well informed on issues of PTRS. Thus, they are believed to have knowledge on the existence of the personal tax relief schemes under the tax law and hence providing the basis for their selection for this current study. The research questionnaire or instruments were largely administered to respondents within their normal working environments.
Out of 236 sets of the research questionnaire distributed, 210 were returned answered and usable thus giving an 88.98% response rate. The data obtained from the completed instruments received were collated and coded into SPSS Version 21 for statistical analysis to enable interpretation.
Attempts were made to administer the questionnaire to all the respondents approached at their various places unless s/he declined to participate in the study. The instrument was largely closed-ended items with possible responses. Closed-ended items were used due to it's ability to obtain the needed data within the shortest possible time line within which respondents were expected to complete the instruments coupled with the number of respondents that the study intended to cover.
The questionnaire for this study consisted of four sections labelled A to D. Section A gathered demographic data, as well as general and experience-related information on the respondents. Section B ascertained the n = N = awareness level of respondents on personal tax relief schemes. Section C focused on the types of personal tax relief schemes and section D attempted to ascertain the factors that influence the usage or otherwise of the personal relief schemes.

RESULTS AND DISCUSSION
The results of the study were analyzed using descriptive statistics and regression model as presented and discussed under the ensuing headings.

Descriptive Statistics
The study revealed that the respondents were made up of 135 (64.30%) males and 75 (35.70%) females (Table 3). This suggests the male taxpayers' dominance in the sample used for this study.
The study respondents' ages ranged between 20 years to above 60 years with the age group 30-39 years dominating as this constituted about 83 respondents (39.50%) of the respondents. The rest were 20 to 29 years with 79 respondents (representing 37.60%), 40 to 49 years with 33 respondents (representing 15.70%), 50 to 59 years with 14 respondents (representing 6.70%), and 60 & above years with 1 respondent (representing 0.50%). It was revealed that the respondent taxpayers have average age of 37 years with standard deviation and coefficient of variation of 37.26 and 1.06 respectively.
Further, the results of the study revealed that the respondents had some form of education as Senior Secondary School Certificate Examination (SSSCE) 13 (6.20%), O' Level 6 (2.90%), A' Level 6 (2.90%), Diploma 54 (25.70%) and degree 126 (60.00%) while those without educational background constituted 5 (2.30%). Those with degree educational background constituted the largest proportion of the study sample, followed by those with a diploma, SSSCE, A' level and O' level in that order suggesting that most respondents have had adequate formal education.
Further, the results of the study revealed that the respondents had working experience ranging from at least one year to above thirty years as shown in Table 3. While those with between 1-5 years working experience dominate the sample studied 106 (50.50%), those with 21 years and  Table 3.
Analysis of the background of the respondents revealed that the sample studied comprised 188 (89.50%) employees, 14 (6.70%) self-employed, 7 (3.30%) unemployed and 1 (0.50%) pensioners as shown in Table 3. This suggested that the sample studied is largely made up of employees and self-employed people representing active taxpayers and hence providing a fertile ground for the intended investigation into the level of awareness and usage of personal tax relief schemes in reducing one's tax burdens or liabilities.

The Level of Awareness and Usage of Personal Income Tax Relief Schemes
The findings of the study on the level of awareness and usage of personal tax relief schemes among the respondents appear in Table 4. The study revealed that 191 (91.00%) of the respondents have ever paid income tax, 14 (6.70%) have never paid income tax, while 5 (2.30%) were not sure if they have ever paid income tax. Probing further, it was found that, out of those who have never paid income tax, are those who earn or receive income that are either exempted from tax or within the marginal relief bracket (specifically within minimum wage band/bracket).
This finding largely coincided with that of [6] observation that, while 91.80% of their study respondents have ever paid tax, about 7.30% never paid. As it were, those who pay income taxes are qualified under the income tax law [14] to apply and use personal tax relief schemes in reducing their taxable base (tax burdens). Consequentially, the study attempted to ascertain the level of awareness and usage of personal tax relief schemes among the respondents and it was revealed that 108 (51.40%) were aware of PTRS, 90 (42.90%) were not aware, while 12 (5.70%) were not sure. This finding is largely contrary to that of [6] study which observed that 78.10% of their study respondents were aware of personal tax relief schemes as against 21.90% who were not aware of the schemes' existence.
The study further revealed that just about 29 out of 210 (representing 14.0%) respondents (that is 29 out of the 108 (27.0%) of those who are aware of PTRS have ever applied and used PTRS to reduce their taxable base as shown in Table 4. This finding is very much consistent with [6] observation and the then Finance and Economic Planning Minister's assertions as reported in the Budget Statement (2012) confirming low usage of PTRS among income taxpayers.
The computed descriptive statistics in Table 4, which suggest fair level (51.40% that is 108 out of 210) of awareness of availability of personal tax relief schemes for reducing tax burdens. Contrary, to the fair level of PTRS awareness, there is however, very low (about 14.00% that is 29 out of 210) usage of PTRS on the part of the respondent income taxpayers studied. This coupled with numerous challenges allegedly encountered in previous attempts could be attributed as the disincentive for the low PTRS usage by the respondent income taxpayers in reducing their tax liabilities as provided for under the income tax laws of Ghana [14].
The challenges identified by the study as being largely responsible for the low usage of personal tax relief schemes among the income taxpayers are shown in Table 4. The study revealed that about 37.93% of the respondents who have ever applied for PTRS alleged that too much time is needed to go through the personal relief application process; 17.60% perceived the attitude of the Tax Officers as uncooperative; 20.69% alleged they were not granted the reliefs when they applied for it in their previous attempts; and 24.14% indicated unavailability of the PTRS application forms at the time of applying as the main challenges encountered. Given this, most respondent income taxpayers may prefer not to use personal tax relief schemes should the above mentioned challenges continue to linger around. From the foregoing, the experiences of the respondent income taxpayers seem to be very much contradictory to the procedure painted under the literature review section. Thus, it is asserted that in applying, taxpayers just need to submit IT Form 21A with necessary information and where applicable supporting documents in respect of reliefs being applied for in case of employee taxpayers. In the case of self-employed, they are to apply together with their returns of income at the end of their basis period. This implies that the authorities at the Ghana Revenue Authority's offices need to review PTRS procedures in order to eliminate the above identified challenges if income taxpayers are to be encouraged to access the personal tax relief schemes as a means of reducing their tax burdens. The citizenry, especially the personal income taxpayers, need to be educated on the required procedures and processes one needs to follow to have an application for personal tax relief approved or granted by the Commissioner General of Ghana Revenue Authority.

ANALYSIS OF REGRESSION RESULTS
The results of the factors likely to influence PTRS usage among Ghanaian individual income taxpayers within the Kumasi Metropolis based on the field data fed into our binary logistics model (as previously stated) are as shown in Tables 5a, 5b, 5c, 5d, 5e, 6 and 7. In all, based on prior studies, 11 factors were listed (as shown in Table  2) to have a significant relationship and/or influence on the respondent income taxpayers' PTRS usage in mitigating tax burdens. Thus, the results of this study (as shown in Table 5) indicate that 181 respondents are predicted as not likely to use PTSR in mitigating their tax burdens at 100% accuracy rate, while 29 respondents are predicted to use PTRS.
The overall classification accuracy rate for the predictive potency of our stated model is at 86.20%, which is seen as pretty good for our purpose. This result is much more like what was exhibited in the frequency distribution table (Table 3). In addition, the variables in the model (i.e. the predictors) are statistically significant in predicting the likelihood of the PTRS usage decisions of the respondents. This gave an assurance to proceed accordingly.
From the result as shown in Table 5c, our stated model as a whole is statistically significant in terms of goodness of fit to better predict judging from a Chi-square value of 60.689 per the omnibus test of model coefficients from Table 5c. This is confirmed by the Model Summary result indicating a Cox & Snell R square value of 0.251 and Nagelkerke R square value of 0.455 and the Hosmer and Lemeshow Test result (Table 5d &  5e); with a Chi-square value of 9.619, which accordingly is not statistically significant confirming the prediction model. Stated differently, the prediction model is able to explain roughly 45.50% of the likelihoods of the PTRS usage by the respondent income taxpayers and should be considered a huge achievement.
However, in term of classifying the respondents into those likely to use PTRS and those not, the overall goodness of fit (per the result as shown in Table 6), revealed that all together, 179 respondents as against 181 (98.90%) were correctly classified as actually not likely to use PTRS.
Also, from Table 6, 15 respondents compared to 29 were correctly confirmed by the model as likely to use PTRS giving 48.30% classification accuracy rate. Overall, this gave a 91.9% accuracy rate in terms of classification. This implies that our model is about 92.0% accurate in predicting the Odds (the probability) of respondents' PTRS usage.

Factors Influencing Individual Personal Income Taxpayers' PTRS Usage
Beyond the goodness of fit of our model in predicting the likelihood of PTRS usage, we now proceed with the individual factors (variables) in the model accordingly as revealed in Table 7. From the result (Table 7 referred), it could be noticed that, even though all the factors (variables) in the model contributed in predicting the Odds (likelihood) of the respondents' PTRS usage in mitigating tax burdens, four of these factors (variables) are however statistically more significant. These are the respondents' knowledge of the tax laws, awareness of PTRS usage in mitigating tax burdens, challenges encountered previously when applying for the PTRS, and perceptions. This finding is consistent with [1] assertions as discussed below.
Based on the binary logistic regression analysis, the respondents' knowledge of the tax laws and its provisions has a negative and significant relationship with PTRS usage for tax burden mitigation as revealed in Table 7 (regression coefficients of -1.171 and Odds ratio of 0.310). Alternatively stated, all things being equal, those with adequate knowledge on the provisions of the tax law in terms of PTRS are more likely to exploit this to manage their tax burdens compared to their counterparts who do not possess adequate knowledge. This implies that, in attempt to have income taxpayers utilize PTRS in mitigating tax burdens, those attempts should be geared towards improving the income taxpayers' knowledge in the tax laws especially those provisions relating to personal income tax reliefs.
Similarly, awareness created as a result of education and/or consciousness has a positive and significant effect on PTRS usage for tax burden mitigation as revealed in Table 7 (regression coefficients of 0.636 and Odds ratio of 1.889). This implies awareness is able to influence the Odds of PTRS usage by respondent taxpayers. Thus, individual personal income taxpayers with fairly good awareness created with education are 1.889 more likely to use PTRS than individual personal income taxpayers without much awareness. As matter of policy for tax authorities, any attempt to have income taxpayers utilize PTRS in mitigating tax burdens and ensure voluntary tax compliance, should be geared towards improving the individual personal income taxpayers' awareness of PTRS usage as enshrined in the tax laws for mitigating personal income tax liabilities.
Challenges encountered in previous attempts to use PTRS are likely to negatively influence the individual personal income taxpayers' PTRS usage as revealed by the regression coefficient of -1.999 and Odds ratio of 0.136 (Table 7 referred). This means that, Challenges encountered in previous attempts  by individual personal income taxpayers in their previous attempts to use PTRS has a negative and significant relationship with their subsequent PTRS usage in tax burden mitigation. Thus, all things being equal, the less or no challenges encountered previously by an income taxpayer, the more likely that income taxpayer is to use PTRS in mitigating his or her tax burdens. As this has consequence for government tax revenue, Tax Authorities and policy makers need to ensure that income taxpayers do not encounter any challenges or bottlenecks when applying for PTRS so as to encourage them subsequently.
From the results, just like challenges encountered in previous attempts in using PTRS, income taxpayers' perceptions also have high likelihood to negatively influence PTRS usage among the respondent income taxpayers as revealed by the regression coefficient of -1.055 and Odds ratio of 0.348 (Table 7 referred). This implies that individual income taxpayers' perception of the procedures for applying for PTRS has negative and significant relationship with PTRS usage for tax burden mitigation. The more favourable the perception is, the more likely the respondent is to utilize PTRS in mitigating tax liabilities.
Thus, unfavourable perception however is more likely to negatively influence income taxpayers' PTRS usage in mitigating tax liabilities.

CONCLUSION AND RECOMMENDA-TIONS
Although this study cannot be said to be without limitations, especially in respect to its inability to cover the whole country as it measures the level of awareness and usage among Ghanaian taxpayers. This may therefore impede the generalization of the inferences made based on the findings. However, the study could be seen as one of the few that lead the way in understanding the provisions of [14] as amended regarding personal tax reliefs schemes and necessary procedures one should follow to access the reliefs. The purpose of this study was to ascertain and/or predict the PTRS awareness and usage level as well as the factors that influence PTRS usage in mitigating tax burdens among individual Ghanaian personal income taxpayers within the Kumasi Metropolis. Also, to impale research interest in personal income tax reliefs schemes considering its effects on government tax revenue and disposable household income and national income in Ghana, an emerging economy.
The results of the study established that though there is fair level of awareness of PTRS, the level of usage however is very low among the respondent personal income tax payers suggesting that personal income tax payers in Kumasi Metropolis are largely not utilizing personal tax relief schemes available under [14] as amended in reducing their taxable base resulting in higher tax liabilities being suffered. Furthermore, the study predicted the respondents' knowledge of the tax laws, awareness of PTRS usage in mitigating tax burdens, challenges encountered previously when applying for the PTRS, and perception as key factors influencing individual personal income taxpayers' PTRS usage within Kumasi Metropolis.
The policy implications stemming from these findings call for the Ghana Revenue Authority, taxpayers and advocates such as the National Civic Educators, as a matter of urgency, to step up educational campaigns on PTRS usage among the various personal income taxpayers especially those with low income so that they can take advantage of the scheme in mitigating their tax burdens if the scheme is actually meant for them. Also, any policy strategy to encourage PTRS should be geared towards influencing these identified factors (and/ or variables) as they significantly influence personal income taxpayers' PTRS usage in mitigating tax burdens. This is critical as PTRS usage has implications for government tax revenue.