The Uncertainty Principle of Game Theory

Integre Technical Publishing Co., Inc.

American Mathematical Monthly 114:8

June 15, 2007

9:48 a.m.

szekely.tex

The Uncertainty Principle of Game Theory Gábor J. Székely and Maria L. Rizzo 1. INTRODUCTION: UNCERTAINTY AND GAMES. If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. —J OHN VON N EUMANN

In 1928, von Neumann’s minimax theorem [33] on two-player, zero-sum games showed that, in general, optimal strategies of players are random, even if the rules of the game do not involve chance. In this paper we investigate the degree of uncertainty or randomness possible in optimal strategies. We show that Heisenberg’s uncertainty principle of physics corresponds to a similar principle for zero-sum games and derive an explicit lower bound for the uncertainty of a game. We also demonstrate that this lower bound is a constraint on the set of possible optimal strategies and apply it to develop a simple but effective method for finding an approximately optimal strategy. A two-player, finite, zero-sum game is traditionally described by a payoff matrix A = (ai j ), where the rows correspond to the strategies of the first player (P) and the columns correspond to the strategies of the second player (Q). (The reason for these particular player “names” will become apparent later.) The individual strategies i and j are called pure strategies. The real number ai j is simply the amount of money or payoff that the first player receives from the second player if the first player chooses strategy “i” and the second player chooses strategy “j”; the chosen strategy of the opponent is unknown to each player. If min max ai j = max min ai j , j

i

i

j

the game represented by A = (ai j ) has a saddlepoint (i ∗ , j ∗ ), and pure strategies i ∗ in arg maxi (min j ai j ) and j ∗ in arg min j (maxi ai j ) are optimal for players P and Q, respectively. (The expression arg min j (·) denotes the set of indices j for which the minimum of the argument is achieved; arg maxi (·) is defined similarly. Although i ∗ and j ∗ are not necessarily unique, one can easily show that ai ∗ j ∗ is unique.) The interesting case is when the game does not have a saddle point. Von Neumann [33] proved that when the game does not have a saddle point, an optimal (minimax) strategy exists for each player but that this optimal strategy is a random (or mixed) strategy. A mixed strategy for the first player is a probability distribution x = (x1 , x2 , . . . , xm ) on the set of pure strategies of the first player. When the game is played, P (randomly) chooses one of his pure strategies according to the probabilities Pr(i) = xi . Similarly, a mixed strategy for the second player is a probability distribution y = (y1 , y2 , . . . , yn ) on the pure strategies of the second player. (N.B. Whenever we use x or y to signify vectors we consider them to be column vectors with m and n coordinates, respectively.) If player P adopts mixed strategy x and player Q adopts mixed strategy y, then the expected payoff is given by the expected value

E (x, y) =

n m

ai j xi y j ,

i=1 j =1

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or, equivalently in matrix notation by E (x, y) = x T Ay, where x T denotes the transpose of x. Player P can find a mixed strategy x ∗ such that his minimum expected gain is at least min y x ∗ T Ay, where the minimum is taken over all distributions y on the pure strategies of Q. Player Q can find a mixed strategy y ∗ such that his maximum expected loss is maxx x T Ay ∗ , where the maximum is taken over all distributions x on the pure strategies of P. It is straightforward that min max x T Ay ≥ max min x T Ay. y

x

x

y

The minimax theorem of von Neumann [33] asserts that in fact min max x T Ay = max min x T Ay. y

x

x

y

(1)

For a proof we refer the reader to von Neumann [33] and von Neumann and Morgenstern [36]. Proofs of the minimax theorem and methods of solving games can also be found in several references (see, for example, [3], [15], [24], [21], or [37]; also, see Owen [24] for linear programming methods of solving games). There are many generalizations of von Neumann’s original minimax theorem: minmax F(x, y) = maxmin F(x, y) is proved for more general sets of x, y, and F than in von Neumann’s theorem (see, for instance, Sion [29] and Groemer [10]; for a simple proof of Sion’s result see Kindler [19]). The common optimal expected payoff v = E (x ∗ , y ∗ ) = x ∗ T Ay ∗ is called the value of the game, and (x ∗ , y ∗ ) is a solution of the game. In this paper we focus on the degree of uncertainty or randomness in optimal mixed strategies. The degree of randomness in optimal mixed strategies depends on the concentration of probability mass on the individual strategies. Note that for mixed strategies it is not meaningful to measure the concentration of probability mass by standard deviation, because the strategies typically are not real numbers. We need a measure of concentration that is applicable for distributions on abstract elements like strategies. Entropy is such a measure. If S denotes the (finite) set of pure strategies of a player and p = ( p1 , p2 , . . . , ps ) is a probability distribution on S, then the entropy H ( p) of p is defined by H ( p) = −

s

pk log2 pk .

(2)

k=1

(We observe the standard convention that 0 log2 0 = 0, which by L’Hospital’s rule is singularity-removing.) It is easily shown that H ( p) ≥ 0 and that H ( p) = 0 if and only if there exists exactly one strategy i such that pi = 1. Entropy is small when p is very concentrated on a few strategies; the discrete uniform distribution on S has maximum entropy log2 s. Remark 1. In the definition of entropy (2), the logarithm base 2 is used rather than the natural logarithm, because one unit of entropy (1 = log2 2) corresponds to the uncertainty of an equally likely yes-no answer. Once we have the answer, this information eliminates one unit of uncertainty. This idea was pioneered by Hartley [11]; in 1946 John Tukey coined the term “bit” (short for binary digit) for this unit, and Shannon [27] first used the word “bit” in print. For recent advances and applications of the notion of entropy the reader can consult [9]. Khinchin [18] is an elegant introduction to the mathematical notion of entropy and its elementary applications. Formula (2) is due to Shannon [27], but the concept and name of entropy originated in the early 1850’s in the work of R. J. E. Clausius. According to Clausius [2, p. 44], October 2007]

THE UNCERTAINTY PRINCIPLE OF GAME THEORY

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Die Energie der Welt ist konstant. Die Entropie der Welt strebt einem maximum zu. That is, one can express a fundamental law of the universe in the following form: The energy of the universe is constant. The entropy of the universe tends to a maximum. While the physical laws of the universe favor maximization of entropy, humans aim to minimize entropy. Humans and all living organisms instinctively try to resist the decay, death, and other sources of increasing entropy by extracting order from their environment. (For more on this idea see Schrödinger [26].) Without human intervention, the tendency toward maximum entropy in the universe would result in erosion, rapid aging, dissolution, etc. On the other hand, if we are able to decrease entropy via increasing information then, at least locally, we gain an advantage for survival. This connection between entropy and information was established in the famous paper of Szilárd [30], which foreshadowed modern cybernetics. (For an English translation see [31].) It is quite natural that a similar principle may apply in game theory. The game theoretic counterpart of Heisenberg’s uncertainty principle is a lower bound for the entropy of optimal strategies of zero-sum games. This lower bound is determined by a single parameter δ = δ(A) of the game, defined by δ=

(min j maxi − maxi min j )ai j , (maxi max j − mini min j )ai j

(3)

if (maxi max j − mini min j )ai j > 0, and δ = 0 otherwise. If a > 0 and b are constants, the optimal strategies of games A = (ai j ) and [a A + b] = (a ai j + b) are identical, so without loss of generality, if δ > 0, we can suppose that maxi, j ai j = 1 and mini, j ai j = 0. Then δ = δ(A) = (min max − max min)ai j j

i

i

j

is determined by the commutator of the nonlinear operators maxi and min j . In the following, δ(A) will denote the value of the commutator (3), or commutator coefficient of A. Our main result is the following: Theorem 1. If G(δ) denotes the class of two-player, finite, zero-sum games with commutator coefficient δ and h(δ) is the entropy of the two-point distribution (1/(1 + δ), δ/(1 + δ)), then a lower bound for entropies of optimal solutions (x ∗ , y ∗ ) of games in G(δ) is given by min(H (x ∗ ), H (y ∗ )) ≥ h(δ). Moreover, h(δ) is the greatest lower bound. The proof of Theorem 1 is given in section 2. First, we consider an example. The Morra game. One of the world’s oldest known games of strategy is the Morra game. It may have originated in Egypt as early as 2000 B . C . and later appeared in 690


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ancient Greece and Rome. In Latin the game is micare digitis, literally “to flash the fingers.” (The name “morra” is a corruption of the verb micare.) A game of quick “flashing fingers” may involve tricks and cheating; in ancient Rome there was an expression for an honest person: “Dignus est quicum in tenebris mices,” which meant “So trustworthy, that one might play Morra with him in the dark.” The game was so common that it was used to settle disputes involving vendors in Roman forums. It appears in Donizetti’s comic opera Deux hommes et une femme as Gasparo and Beppe play to decide who will stay with Rita (and both try to lose). Today the game is actually illegal in Italy, after laws passed in the twentieth century aimed to curb violence in situations where tempers between rivals flared. Nevertheless, this popular game is still played at festas in many parts of southern Italy. In our version of the Morra game, each player shows one, two, or three fingers, and simultaneously each calls his guess of the number of fingers his opponent will show. If both players guess correctly, or both guess incorrectly, the game is a draw. If exactly one player guesses correctly, he wins an amount equal to the sum of the fingers shown by both players. This example, which we might call the three-finger Morra game, appears in [3] and [4]. Good [8] discusses the f -finger generalization of the Morra game. The strategies for each player are pairs (d, g), where d is the number of fingers and g is the guess. Thus, each player has nine pure strategies: (1, 1), (1, 2), . . . , (3, 3). The payoff matrix is given in Table 1. Table 1. Payoff matrix of the game of Morra.

Strategy 1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

0 −2 −2 3 0 0 4 0 0

2 0 0 0 −3 −3 4 0 0

2 0 0 3 0 0 0 −4 −4

−3 0 −3 0 4 0 0 5 0

0 3 0 −4 0 −4 0 5 0

0 3 0 0 4 0 −5 0 −5

−4 −4 0 0 0 5 0 0 6

0 0 4 −5 −5 0 0 0 6

0 0 4 0 0 5 −6 −6 0

Denote the payoff matrix by A = (ai j ). Then (min max − max min)ai j = min max ai j − max min ai j = 3 − (−3) > 0, j

i

i

j

j

i

i

j

so by the minimax theorem optimal strategies for both players are mixed strategies. If a player adopts one of his optimal strategies (by “symmetry” they are the same for both players), his expected payoff is at least v = 0, the value of the game. If his opponent also adopts an optimal strategy, then both players receive expected payoff v = 0. Hence, if both players adopt optimal strategies, then all optimal strategies are equally good. However, not all optimal strategies are equally random. The game can be solved by elimination of dominated strategies and considering square submatrices (see, for example, [3], [4], or [24]). The optimal strategies for a player of a zero-sum game form a convex set determined by its extreme points [28]. It can be shown (as in [3] or [4]) that the extreme points of the set of optimal strategies for this Morra game are October 2007]


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x ∗ = y ∗ = (0, 0, 5/12, 0, 4/12, 0, 3/12, 0, 0),

(4)

x ∗ = y ∗ = (0, 0, 16/37, 0, 12/37, 0, 9/37, 0, 0),

(5)

∗

∗

x = y = (0, 0, 20/47, 0, 15/47, 0, 12/47, 0, 0),

(6)

x ∗ = y ∗ = (0, 0, 25/61, 0, 20/61, 0, 16/61, 0, 0).

(7)

Note that although all of these mixed strategies are optimal, their entropies are not identical. The entropies of solutions (4) – (7) are 1.554585, 1.545970, 1.553281, and 1.561312, respectively. The commutator coefficient of the Morra game is δ(A) =

1 3 − (−3) min j maxi ai j − maxi min j ai j = . = maxi max j ai j − mini min j ai j 6 − (−6) 2

Theorem 1 implies that for all m × n games A such that δ(A) = 1/2, entropies of . optimal solutions are bounded below by h(1/2) = 0.9182958, which is the entropy of the two point distribution (2/3, 1/3). Heisenberg and von Neumann. Let X and Y denote the set of all probability distributions on the pure strategies of P and Q, respectively. The actions of players P and Q can be described in terms of a maximum operator P and a minimum operator Q acting on E as follows. Let φ(x, y) = φ(x1 , x2 , . . . , xm , y1 , y2 , . . . , yn ) be a function defined on a compact set in Rm+n . The operator P acting on φ takes the maximum value of φ with respect to x for each fixed y, and therefore maps φ to a function of n variables, the coordinates of y. Similarly, Qφ records the minimum value of φ with respect to y, and therefore maps φ to a function of x. For each game the expectation E is a function on the compact set X × Y, and (Q P)E = min max E (x, y), y∈Y x∈X

(P Q)E = max min E (x, y). x∈X y∈Y

The commutator of operators P, Q applied to E is therefore [P, Q]E = (Q P − P Q)E = min max − max min E (x, y). y∈Y x∈X

x∈X y∈Y

Von Neumann’s minimax theorem can be reformulated as follows: for every finite zero-sum game (and thus for every E ) it is true that [P, Q]E = 0. In other words, there always exist optimal minimax strategies x ∗ and y ∗ such that (P Q)E = (Q P)E = E (x ∗ , y ∗ ) = v. Our Theorem 1 states that there is a lower bound on the randomness of optimal strategies of P and Q in terms of the commutator coefficient δ = δ(A). Thus, depending on δ, the distributions x ∗ and y ∗ cannot be too concentrated. In his celebrated 1927 paper, Heisenberg [12] formulated the uncertainty principle (for an English translation see [38, pp. 62–84]). In quantum mechanics, the measurements of the position (P) and the momentum (Q) of an electron are random variables. 692


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Both P and Q for a system in a given state vary according to certain probability distributions. The uncertainty principle states that the more precisely the position of an electron is determined, the less precisely the momentum is known, and vice versa. This was expressed by Heisenberg as ε(P)ε(Q) ≈ ,

(8)

where ε(P) and ε(Q) denote the imprecisions or errors of the position and momentum, respectively, and = h/(2π), where h is the Planck constant. In the reformulation by von Neumann [34], particles, like an electron with one degree of freedom, can be represented as complex-valued state functions f on the real line R, f : R → C. Here | f (x)|2 is the probability density of the particle on R, and therefore the (integral) scalar product satisfies ( f, f ) = 1. Let L 2 (R) denote the Hilbert space of complex-valued square integrable functions on the real line. The observables, like P and Q, can be represented as self-adjoint linear operators: (Q f )(x) = x f (x),

(P f )(x) = −i f (x),

where the domain of Q is { f ∈ L 2 (R) : x f (x) ∈ L 2 (R)}, the domain of P is the set of all absolutely continuous functions f in L 2 (R) such that f (x) belongs to L 2 (R), and i is the complex unit. If [P, Q] denotes the commutator of P and Q, then [P, Q] f = (Q P − P Q) f = i f

(9)

for all f for which the left-hand side is defined; this is called the canonical commutation relation. The absolute value of ([P, Q] f, f ), evaluated using (9), is equal to the right-hand side of (8). The standard deviation (A, f ) of an arbitrary self-adjoint operator A (“observable”) acting on f is defined by (A, f ) = ((A2 f, f ) − (A f, f )2 )1/2 , analogous to the standard deviation of a random variable in terms of its first and second moments. Kennard [17] showed that (P, f )(Q, f ) ≥

1 . 2

Thus the probability distributions corresponding to P and Q cannot be too concentrated at the same time. In general, according to Robertson [25], for any pair of observables A and B we have (A, f )(B, f ) ≥

1 |( f, [A, B] f )|. 2

Now let us interpret games in the context of quantum physics. In the theory of zero-sum games, the “state” can be represented by a payoff matrix f = [ f (i, j)]. If (maxi max j − mini min j ) f (i, j) > 0, f can be normalized so that max max f (i, j) = 1 i

j

and

min min f (i, j) = 0. i

j

The value of the commutator δ = (P Q − Q P) f is the game theoretic counterpart of i in (9): October 2007]


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δ = [P, Q] f = (Q P − P Q) f = (min max − max min) f (i, j). j

i

i

j

Following this analogy, the randomness (entropy) of optimal mixed strategies is bounded below by a function of δ. Theorem 1 shows that this bound indeed exists: when δ > 0, the greatest lower bound is exactly h(δ), the entropy of the two point distribution (1/(1 + δ), δ/(1 + δ)). This is what we call the uncertainty principle of game theory. In quantum physics the operators P and Q are linear operators. The operators maxi and min j are obviously not linear: typically maxi [ f (i, j) + g(i, j)] = maxi f (i, j) + maxi g(i, j). Thus, in game theory we have an example of an uncertainty principle determined by the commutator of nonlinear operators that is analogous to the uncertainty principle of physics. (Note that P is subadditive and Q superadditive, and both preserve multiplication by nonnegative constants, but we are not aware of any generalization of Heisenberg’s uncertainty principle based on the commutator of such operators.) Heisenberg’s uncertainty principle also has an entropic form in which the uncertainty of the position and the momentum of a particle are measured by their entropies. Since these are continuous quantities, the definition of entropy (2) is modified. If | f |2 is a probability density function of the location of a particle, then its entropy is H (| f |2 ) = −

| f (x)|2 log2 | f (x)|2 d x.

Interestingly, unlike the discrete case, H (| f |2 ) can take negative values: even H (| f |2 ) = −∞ is possible if | f |2 is not bounded, which corresponds to a high degree of concentration of the probability. The entropic form of Heisenberg’s uncertainty principle is due to Hirschman [14] and Beckner [1] (see also Kraus [20] and Hilgevoord and Uffink [13]). If the state function of the position is f , then the momentum function is the Fourier transform 1 (F f )(x) = √ 2π

f (t) exp

−it x dt.

In the following suppose for simplicity that = 1. The density function of the momentum is |F f |2 and the L 2 -norm of f is f 2 . The entropic form of the uncertainty principle states that if f is in L 2 (R) and || f ||2 = 1, then H (| f |2 ) + H (|F f |2 ) ≥ log2 (πe), provided that the left-hand side is defined. Note that unlike the products of standard deviations in the original version of Heisenberg’s uncertainty principle, here we have sums of entropies. For generalization to two arbitrary self-adjoint operators we refer the reader to Maasen and Uffink [23], Maasen [22], and Uffink [32]. Here we have arrived at an important meta-theorem in harmonic analysis, which states that a nonzero function and its Fourier transform cannot both be sharply localized. On the history of this problem and many recent related results there is a very interesting paper by Folland and Sitaram [6]. 2. ENTROPY OF OPTIMAL MIXED STRATEGIES. Our proof of Theorem 1 begins with a lemma for the 2 × 2 case. In the following, matrices are delimited with square brackets. 694


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Lemma 1. Let A be a 2 × 2 zero-sum game with commutator coefficient δ = δ(A). If (x ∗ , y ∗ ) is a solution of game A, then min(H (x ∗ ), H (y ∗ )) ≥ h(δ), and h(δ) is the greatest lower bound. Proof. Consider the finite, two-player, zero-sum game represented by the payoff matrix a11 a12 . A= a21 a22 If a11 = a12 = a21 = a22 (that is, A is a trivial game) then δ(A) = 0, and min(H (x ∗ ), H (y ∗ )) ≥ 0 = h(δ) for every pair of optimal strategies (x ∗ , y ∗ ). Suppose that A is a nontrivial game. Without loss of generality, we may assume that maxi max j ai j = 1 and mini min j ai j = 0. In this case the entries of A can be more conveniently denoted by the four numbers 0, a, b and 1, where 0 ≤ a ≤ b ≤ 1. Clearly if the diagonal of A contains both 1 and 0, there is a saddle point and δ(A) = 0. In this case each player has optimal pure strategies, for which H (x ∗ ) = H (y ∗ ) = 0 = h(δ). If a = b the game has a saddle point, δ(A) = b − a = 0, and H (x ∗ ) = H (y ∗ ) = 0 = h(δ). If optimal solutions of A are mixed strategies, we have 0 ≤ a < b ≤ 1, and the diagonal cannot contain both 1 and 0. In this case, the payoff matrix (or its transpose) is of the form 1 0 A= (10) a b with 0 ≤ a < b ≤ 1, a = maxi min j ai j , b = min j maxi ai j , and δ = b − a > 0. Here, the unique solution of game A is given by x ∗T = [1 1]A−1 v,

y ∗ = A−1 [1 1]T v,

where A−1 is the inverse of A, and v = 1/{[1 1]A−1 [1 1]T } = b/(1 + b − a) is the value of the game (see, for example, [24, p. 27]). Hence

x

∗T

b−a = 1+b−a

1 δ = 1+b−a 1+δ

1 , 1+δ

and H (x ∗ ) = h(δ). For the second player we have

y

∗T

b = 1+b−a

1−a δ+a = 1+b−a 1+δ

1−a . 1+δ

Regarding (a, δ) rather than (a, b) as the parameters defining A, and fixing δ > 0, we see that H (y ∗ ) = H (y ∗ (a)) is a continuously differentiable function of a (0 < a < 1 − δ). By elementary calculus we obtain October 2007]


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1 d H (y ∗ (a)) =− log2 da 1+δ

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.

Therefore H (y ∗ (a)) is strictly increasing on the interval (0, 1 − b) and strictly decreasing on (1 − b, 1 − δ), so the maximum occurs when a + b = 1. The minimum of H (y ∗ (a)) for fixed δ with 0 < δ ≤ 1 occurs when a = 0 or a = 1 − δ. If a = 0, y ∗ T = [δ/(1 + δ) 1/(1 + δ)]

and

H (y ∗ ) = h(δ);

if a = 1 − δ, y ∗ T = [1/(1 + δ)

δ/(1 + δ)]

and again H (y ∗ ) = h(δ). Thus, for all possible values of a we have H (y ∗ ) ≥ h(δ), and min(H (x ∗ ), H (y ∗ )) = min(h(δ), H (y ∗ )) = h(δ). Considering the transpose of (10) we obtain the same result with the labels of players P and Q reversed. Notice that for game (10) with δ > 0 the minimum h(δ) is achieved when the optimal solutions are the two point distributions (δ/(1 + δ), 1/(1 + δ)) or (1/(1 + δ), δ/(1 + δ)). As δ → 0 we see that the corresponding solutions approach a pure strategy solution, with entropy zero. As δ → 1 the solutions corresponding to minimum entropy have limit (1/2, 1/2) with entropy one. Remark 2. The proof of Lemma 1 actually establishes a stronger statement. That is, for all nontrivial 2 × 2 games A the entropies of solutions (x ∗ , y ∗ ) satisfy min(H (x ∗ ), H (y ∗ )) = h(δ(A)). In general, optimal mixed strategies may not necessarily exclude the possibility that optimal pure strategies exist. For simplicity, we consider the basic solutions as defined by Shapley and Snow [28]. If Z is a bounded, nonempty, closed, and convex subset of Rn , define Z ∗ to be the smallest subset of Z that spans Z; that is, Z ∗ is the smallest set whose convex hull is Z. Let X and Y denote the sets of optimal strategies x ∗ and y ∗ , respectively, for game A. Then (x ∗ , y ∗ ) is a basic solution of A if x ∗ belongs to X ∗ and y ∗ to Y ∗ . The basic solutions determine the convex set containing all optimal solutions. Thus, whenever δ(A) > 0, the basic solutions of A are mixed strategy solutions. When a game has a basic mixed strategy solution, no pure strategy is optimal. Proof of Theorem 1. Suppose that x ∗ and y ∗ are optimal strategies of an m × n game A in G(δ). We need to prove that min(H (x ∗ ), H (y ∗ )) ≥ h(δ)

(11)

and that h(δ) is the greatest lower bound. The case m = n = 2 is proved by Lemma 1. Since entropy is nonnegative, statement (11) is also clearly true whenever the game has a saddle point. 696


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By Theorem 2 of Shapley and Snow [28], a necessary and sufficient condition that (x ∗ , y ∗ ) be a basic solution of a nonsquare game A is that there exists a nonsingular square submatrix M of A such that x ∗ and y ∗ are optimal for M. If ξ ∗ and η∗ are basic optimal mixed strategies for P and x ∗ is a convex combination of ξ ∗ and η∗ , then H (x ∗ ) ≥ min(H (ξ ∗ ), H (η∗ )). Therefore, we need only prove that (11) holds for basic mixed strategy solutions of square m × m games A with m > 2. Let x = (x1 , x2 , . . . , xm ) be a probability distribution, where 0 ≤ x1 ≤ x2 · · · ≤ xm < 1 and m > 2. Fix the maximum value xm of x. Then the distribution x can be expressed as (t1 xm , t2 xm , . . . , tm−2 xm , 1 − sxm , xm ), where ti = xi /xm and s = 1 + t1 + · · · + tm−2 . Thus, for given xm , the entropy H (x) is a function of t = (t1 , . . . , tm−2 ), which we denote H (t; xm ), and H (t; xm ) = −

m−2

ti xm log2 (ti xm ) − (1 − sxm ) log2 (1 − sxm ) − xm log2 xm .

i=1

Note that xm−1 > 0 and 1 − sxm > 0 when 0 < xm < 1. If ti = 0, then according to the definition of H (·) we have ti xm log2 ti xm = 0. For i = 1, 2, . . . , m − 2 such that ti > 0, the partial derivatives of H (t; xm ) are ∂ H (t; xm ) 1 (−xm log ti xm − xm + xm log(1 − sxm ) + xm ) = ∂ti log 2 1 − sxm xm−1 = xm log2 = xm log2 ≥ 0. ti xm xi The partial derivatives of H (t; xm ) satisfy ∂ H (t; xm ) > 0, ∂ti

if 0 < xi < xm−1 ,

∂ H (t; xm ) = 0, ∂ti

if xi = xm−1 or xi = 0;

hence H (t; xm ) is minimized when t1 = · · · = tm−2 = 0. Therefore, entropy for probability distributions x such that maxi xi = xm is bounded below by H (0, 0, . . . , 0, 1 − xm , xm ) = H (1 − xm , xm ). Now suppose that x ∗ is a basic optimal mixed strategy for player P of an m × m ∗ game A with δ = δ(A) > 0. In this case 0 < xm−1 ≤ xm∗ < 1, and H (x ∗ ) is bounded ∗ ∗ below by H (1 − xm , xm ). Applying constraints satisfied in the 2 × 2 case we conclude that xm∗ ≤ 1/(1 + δ), and that H (1 − xm∗ , xm∗ ) is a decreasing function of the maximum probability xm∗ . It follows that

δ 1 , H (x ) ≥ H 1+δ 1+δ ∗

= h(δ).

Similarly, if y ∗ is a basic optimal mixed strategy for player Q, then H (y ∗ ) ≥ h(δ), and min(H (x ∗ ), H (y ∗ )) ≥ h(δ). The following example proves that the lower bound h(δ) is sharp (i.e., for each m (m ≥ 2) and n (n ≥ 2) the lower bound h(δ) of entropy of solutions is achieved for some m × n game with positive commutator coefficient δ). Define a game Am,m (δ) with square payoff matrix that is upper triangular with entries ai j = 1, when j ≥ i, October 2007]


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except that a12 = 1 − δ. For example, the 5 × 5 game A5,5 (δ) has payoff matrix ⎤ ⎡ 1 1−δ 1 1 1 1 1 1 1 ⎥ ⎢ 0 ⎥ ⎢ 0 1 1 1 ⎥. A5,5 (δ) = ⎢ 0 ⎣ 0 0 0 1 1 ⎦ 0 0 0 0 1 By elimination of dominated strategies we can easily obtain a basic solution of Am,m (δ) whenever m ≥ 2 and 0 < δ < 1, which is T T 1 δ δ 1 0 0 . . . 0 , y∗ = 0 0 . . . 0 , (12) x∗ = 1+δ 1+δ 1+δ 1+δ and min(H (x ∗ ), H (y ∗ )) = h(δ). If n > m ≥ 2 define the game Am,n (δ) by augmenting the square matrix Am,m (δ) with n − m columns of ones. Then Am,n (δ) has commutator coefficient δ and solution vectors of the form (12), for which the lower bound h(δ) of minimum entropy is attained. If m > n ≥ 2, consider the game represented by the transpose of An,m (δ). Remark 3. It is also true that entropies of optimal strategies are bounded above. The upper bound depends on the dimension of the payoff matrix. Optimal strategies x ∗ and y ∗ of an m × n game are each supported on at most k = min(m, n) strategies, so 1 1 ∗ ∗ log2 max(H (x ), H (y )) ≤ −k = log2 (min(m, n)). k k 3. MINIMUM ENTROPY STRATEGIES. In many situations, although a zerosum game theoretically can be solved, the computation of optimal strategies is beyond the current state of the art in available software and hardware. The number of strategies might be too large, the system might be numerically ill-conditioned, or there might be a need to act in real time or almost real time. In this case, we seek some elementary method of analyzing the game that provides a good strategy in a reasonably short time given limited resources. Methods do exist for computing numerical approximations to solutions of games. Here, however, we take a different and simpler approach. We suppose that a solution is not readily available and seek an alternate mixed strategy that is good in the sense that the expected payoff is above a certain lower bound. We analyze the game from the perspective of player P. Assume that the commutator coefficient of the game is positive. Then P realizes that optimal strategies are mixed strategies. Among possible mixed strategies, the simplest to analyze and implement are the two-point strategies. The objective function to be optimized is the expected payoff x T Ay. However, in order for player P to compute his expected payoff for any proposed strategy x, the strategy y of the opponent must be specified. By the minimax theorem one should assume that the opponent adopts an optimal mixed strategy, but in this situation the optimal strategy is unknown to P. Therefore, in order to evaluate the expected payoff of any proposed mixed strategy, it is necessary to assign some probability distribution to the opponent’s actions as a substitute for his (unknown) optimal strategy. Some guidance is provided by the interpretation of the entropy of a probability distribution. The entropy is small when the probability mass is very concentrated on a few strategies. Minimum entropy corresponds to the entropy h(δ) of the two-point distribution (1/(1 + δ), δ/(1 + δ)), where δ is the commutator coefficient of the game. 698


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As explained in what follows, P should consider the two-point minimum entropy distributions of both players. The idea of minimum entropy is partly motivated by the explanation of the minimax theorem: player P seeks to maximize his minimum expected gain, while player Q seeks to minimize his maximum expected loss. Thus each player is considering the least unfavorable outcome. The set of minimum entropy distributions consists of the most extreme two point distributions that could possibly be optimal. We apply the minimax principle to the two-point minimum entropy distributions. The lower bound h(δ) on entropy implies a constraint on the probability distributions of optimal strategies of both players, such that any strategy is assigned probability at most 1/(1 + δ). Let a(i1) ≤ a(i2) ≤ · · · ≤ a(in) list the ordered payoffs of row i in game A. Given that Q adopts an optimal strategy in game A, P can expect at least a(i1)

1 δ + a(i2) 1+δ 1+δ

against his pure strategy i. This establishes a set of lower bounds on P’s expected payoff for each pure strategy i against an optimal strategy of Q. Therefore P should prefer a pure strategy i ∗∗ that attains the greatest lower bound, and assigns probability 1/(1 + δ) to i ∗∗ . Let x(i, k; δ) (i = k), denote the two-point probability distribution given by ⎧ 1 , r = i; ⎪ ⎨ 1+δ δ x(i, k; δ)r = , r = k; ⎪ ⎩ 1+δ 0, r∈ / {i, k}. For every pair of strategies (i, k) of player P compute the minimum expected payoff of strategy x(i, k; δ) against all two-point strategies y( j, l; δ) of player Q. Finally compute the maximum of these minimums (the maxmin) over all pairs (i, k). Thus the proposed strategy for player P on game A with commutator coefficient δ = δ(A) is x(i ∗∗ , k ∗∗ ; δ), where (i ∗∗ , k ∗∗ ) ∈ arg max min x(i, k; δ)T Ay( j, l; δ). i =k

j =l

(13)

If (i ∗∗ , k ∗∗ ) is not unique, randomly choose one pair that satisfies (13). Then x(i ∗∗ , k ∗∗ ; δ)T Ay ∗ is the expected value P receives if P adopts the strategy x(i ∗∗ , k ∗∗ ; δ) against an optimal strategy y ∗ of Q. Strategy x(i ∗∗ , k ∗∗ ; δ) is not an estimate of a solution; it is an alternate strategy that is “optimal” in a different sense. Note that (i ∗∗ , k ∗∗ ) is determined by the two smallest payoffs in each row, so that in general finding (i ∗∗ , k ∗∗ ) requires much less computational effort than solving the game. The Morra game. To illustrate the method, we find a minimum entropy strategy for the Morra game described in section 1, for which δ = 1/2. The payoff matrix in Table 1 is equivalent to one with maximum payoff 1, minimum payoff 0, and value v = 0.5. The minimum entropy pair (i ∗∗ , k ∗∗ ) is (2, 6), and the proposed two-point mixed strategy is x ∗∗ = [0 2/3 0 0 0 1/3 0 0 0]T . If player P adopts strategy x ∗∗ against any of the optimal strategies (4)–(7) of Q, the expected payoff v ∗∗ = x ∗∗ T Ay ∗ is at least 0.496. Compared with the optimal value October 2007]


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v = 0.5, this simple approach to finding a good two-point strategy has a relative error of 0.008. Thus, without solving the game or having any knowledge of the optimal strategies of the opponent, P found a two-point mixed strategy that is within one percent of optimal. Interestingly, the support set of x ∗∗ is actually disjoint from the support set of optimal strategies (4)–(7). Random payoff matrix. We applied the minimum entropy approach to square games with random payoffs in a simulation experiment. The payoffs of each game were randomly generated from the continuous Uniform(0,1) distribution. The simulation experiment was designed to estimate the expected payoff that player P would receive if he adopted a minimum entropy strategy x ∗∗ against an optimal strategy y ∗ of Q. The expected payoff v ∗∗ = x ∗∗ T Ay ∗ of the minimum entropy strategy from (13) was then compared with the optimal value of the game v = x ∗ T Ay ∗ obtained by standard linear programming methods (see, for example, [16], [24] or [37]). In large randomly generated payoff matrices, saddle points rarely occur [5], [7]; in this simulation we simply discarded any games with saddle points, so all of the games analyzed have mixed strategy solutions. Table 2 summarizes the distribution of relative error (v − v ∗∗ )/v for 10000 simulated mixed strategy 10 × 10 games. The relative error was (essentially) zero in 83 percent of the games, and less than 0.1 in 98 percent of the games. For comparison purposes, the results are also shown for the distribution of relative error when player P randomly selects any pure strategy with equal probability. Table 2. Relative error (v − v ∗∗ )/v for 10000 randomly generated 10 × 10 games.

Percentile

Min. Entropy Error

Random Strategy Error

83 85 90 95 98 99

< 1.7e−14 0.008 0.028 0.062 0.098 0.123

0.251 0.264 0.309 0.386 0.454 0.554

4. SUMMARY. The lower bound for entropy of optimal strategies of zero-sum games is the natural game theoretic counterpart of the lower bound of the uncertainty principle of physics. In quantum physics the lower bound for randomness in measurements is given in terms of a commutator of two linear operators that describe “observables”. In zero-sum games the lower bound for randomness of optimal solutions is given in terms of the value δ of the commutator of two nonlinear operators on the payoff matrix, maximum and minimum. In case δ > 0, minimal entropy of optimal mixed strategy solutions is achieved when the strategy is supported on exactly two points with probabilities 1/(1 + δ), δ/(1 + δ). This result holds for m × n games, for integers m ≥ 2 and n ≥ 2. Thus, regardless of the number of possible strategies of a game, the lower bound of entropy is determined by a single parameter δ. Our uncertainty principle suggests searching the corresponding minimum entropy distributions to determine a good two point strategy if a solution is not available. This simple two point minimum entropy estimate is easily computed and quite effective in our examples. 700


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DEDICATION The authors dedicate this paper to the memory of John von Neumann (1903–1957).

ACKNOWLEDGMENT The authors thank three anonymous referees for many suggestions that greatly improved the paper.

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37. N. N. Vorob’ev, Game Theory, Lectures for Economists and System Scientists (trans. S. Kotz, with supplement), Springer-Verlag, New York, 1977. 38. J. A. Wheeler and H. Zurek, Quantum Theory and Measurement, Princeton University Press, Princeton, 1983. ´ ´ GABOR J. SZEKELY received his Ph.D. from ELTE, Budapest, and the Doctor of Science degree from the Hungarian Academy of Sciences. He is professor at Bowling Green State University and senior researcher of the Rényi Institute of the Hungarian Academy of Sciences. Between 1985 and 1995 he was the program manager of the Budapest Semesters in Mathematics. Székely is a past chair of the Department of Stochastics of the Budapest Institute of Technology and recipient of the Rollo Davidson Prize (Cambridge). His publications include the Monographs Paradoxes in Probability Theory and Statistics (Reidel, 1986) and Algebraic Probability Theory (with I.Z. Ruzsa, Wiley, 1988). He likes to adopt principles of physics into mathematics and statistics; besides this paper, examples include his E -statistics (energy statistics), implemented in the energy package for R. Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403 [email protected]

MARIA L. RIZZO received her Ph.D. from Bowling Green State University, was assistant professor in the Department of Mathematics at Ohio University for four years, and presently is assistant professor at Bowling Green State University. Her research interests include multivariate analysis, computational statistics, classification, goodness-of-fit tests, and especially all applications of energy statistics. Related interests include software development, such as the energy package for R, and statistical consulting. Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403 [email protected]

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