April 25 May 6, 2016, Verona, Italy. GAME THEORY and APPLICATIONS Mikhail Ivanov Krastanov

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1 April 25 May 6, 2016, Verona, Italy GAME THEORY and APPLICATIONS Mikhail Ivanov Krastanov

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3 Games in normal form There are given n-players. The set of all strategies (possible actions) of the i-th player is denoted by Σ i. Each player choose an element σ i Σ i, i = 1, 2,..., n. Then the mathematical expectation of the payo function for the i-th player is given by π i (σ 1, σ 2,..., σ n ), i = 1, 2,..., n. Denition We say that the sets Σ i, i = 1, 2,..., n, and the functions π i : Σ 1, Σ 2 Σ n R, i = 1, 2,..., n, determine a game Γ in normal form. Remark Each player want to maximize his payo function, but this depends on the choice of all players!

4 Nash equilibriom Denition Let the game Γ is determined by the sets Σ i, i = 1, 2,..., n, and the functions π i : Σ 1, Σ 2 Σ n R, i = 1, 2,..., n. It is said that the n-tuple Denition We say that the sets Σ i, i = 1, 2,..., n, and the functions π i : Σ 1, Σ 2 Σ n R, i = 1, 2,..., n, determine a game Γ in normal form. Remark Each player want to maximize his payo function, but this depends on the choice of all players!

5 Nash equilibrium Denition Given a game Γ, a strategy n-tuple ( σ 1, σ 2,..., σ n ) is said to be a Nash equilibrium if for any index i = 1, 2,..., n, and any σ i Σ i, π i ( σ 1,..., σ i 1, σ i, σ i+1..., σ n ) π( σ 1, σ 2,..., σ n ). In other words, a strategy n-tuple ( σ 1, σ 2,..., σ n ) is said to be a Nash equilibrium if no player has a reason to change his strategy, assuming that none of the other players is going to change his strategy.

6 Zero-sum games Denition A game Γ is said to be zero-sum if Σ n π i (σ 1,..., σ i 1, σ i, σ i+1..., σ n ) = 0 for any σ i Σ i, i = 1,..., n. In general, a zero-sum games represents a closed system: everything that someone wins must be lost by someone else. Most parlor games are of the zero-sum type. Two-person zero-sum games are called strictly competitive games.

7 Let ( σ 1, σ 2 ) Σ 1 Σ 2 be a Nash equilibrium. Then π 1 (σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) for each σ 1 Σ 1 and π 2 ( σ 1, σ 2 ) π 2 ( σ 1, σ 2 ) for each σ 2 Σ 2. Because in this case π 2 (σ 1, σ 2 ) = π 1 (σ 1, σ 2 ), we obtain that π 1 (σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) for each σ 1 Σ 1 and each σ 2 Σ 2, i.e. ( σ 1, σ 2 ) is a saddle point for the function π 1.

8 We have obtained that the rst player want to maximize π 1 (σ 1, σ 2 ) over Σ 1, while the second player want to minimize π 1 ( σ 1, σ 2 ) over Σ 2. Let us assume that rst player is omniscient and he can guess correctly the action σ 2 Σ 2 of the second player. Then the rst player will choose σ 1 Σ 1 so that π 1 ( σ 1, σ 2 ) = max σ 1 Σ 1 π 1 (σ 1, σ 2 ). For simplicity, we assume here and further that these max and min exist. But then it is natural the second player to choose σ 2 Σ 2 so that ( ) max π 1 (σ 1, σ 2 ) = min σ 1 Σ 1 σ 2 Σ 2 max π 1 (σ 1, σ 2 ) σ 1 Σ 1

9 Analogously it is natural for the rst player to choose σ 1 Σ 1 so that ( ) Denition It is natural to dene min π 1 ( σ 1, σ 2 ) = max σ 2 Σ 2 σ 1 Σ 1 gain-oor v I := max σ 1 Σ 1 ( min π 1 (σ 1, σ 2 ) σ 2 Σ 2 ) min π 1 (σ 1, σ 2 ) σ 2 Σ 2 and loss-ceiling v II := min σ 2 Σ 2 ( ) max π 1 (σ 1, σ 2 ). σ 1 Σ 1 The meaning of v I and v II is: the rst player should not win less than v I and the second player should not loss more than v II.

10 Lemma 1. We have that v I v II. Proof: Let us x arbitrary elements ˆσ 1 Σ 1 and ˆσ 2 Σ 2. Clearly, we have that max π 1 (σ 1, ˆσ 2 ) π 1 (ˆσ 1, ˆσ 2 ) min π 1 (ˆσ 1, σ 2 ), i.e. σ 1 Σ 1 σ 2 Σ 2 max π 1 (σ 1, ˆσ 2 ) min π 1 (ˆσ 1, σ 2 ), σ 1 Σ 1 σ 2 Σ 2 Since the right-hand side does not depend on σ 1 and the left-hand side does not depend on σ 2, we obtain that

11 Proof: (continuation) ( min σ 2 Σ 2 max π 1 (σ 1, ˆσ 2 ) σ 1 Σ 1 ) max σ 1 Σ 1 i.e. v II v I. This completes the proof. Denition. ( ) min π 1 (ˆσ 1, σ 2 ), σ 2 Σ 2 If v I = v II, the common number v := v I = v II is said to be value of the game Γ.

12 Lemma 2. A game Γ has a value, if and only if, it has a saddle point. Proof. (suciency) Let the game Γ has a saddle point, i.e. there exists ( σ 1, σ 2 ) Σ 1 Σ 2 so that π 1 (σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) for each σ 1 Σ 1 and each σ 2 Σ 2. Then and hence max σ 1 Σ 1 π 1 (σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) min σ 2 Σ 2 π 1 ( σ 1, σ 2 )

13 Proof. (suciency: continuation) ( ) min σ 2 Σ 2 max π 1 (σ 1, σ 2 ) σ 1 Σ 1 max σ 1 Σ 1 ( ) min π 1 ( σ 1, σ 2 ) σ 2 Σ 2 i.e. v II v I. We have already prove that v I v II. Hence v I = v II and the game Γ has a value. This completes the proof of the suciency. Proof. (necessity) Let the game Γ has a value ( v = min σ 2 Σ 2 max π 1 (σ 1, σ 2 ) σ 1 Σ 1 ) max σ 1 Σ 1 ( ) min π 1 ( σ 1, σ 2 ). σ 2 Σ 2

14 Proof. (necessity: continuation) According our simplifying assumption, there exist σ 1 Σ 1 and σ 2 Σ 2 so that and hence v = min σ 2 Σ 2 π 1 ( σ 1, σ 2 ) = max σ 1 Σ 1 π 1 (σ 1, σ 2 ), π 1 (σ 1, σ 2 ) v π 1 ( σ 1, σ 2 ) for each σ 1 Σ 1 and each σ 2 Σ 2. It follows from here that π 1 (σ 1, σ 2 ) v = π 1 ( σ 1, σ 2 ) π 1 ( σ 1, σ 2 ) for each σ 1 Σ 1 and each σ 2 Σ 2, i.e. ( σ 1, σ 2 ) is a saddle point for the function π 1. This completes the proof of the necessity.

15 Denition A mixed strategy is a probability distribution on the set of his pure strategies. Let the rst player has m pure strategies. Then the set X of its mixed strategies consists of all vectors x = (x 1, x 2,..., x m ) whose components satisfy x i = 1 and x i 0, i = 1,..., m. Analogously, if the second player has n pure strategies. Then the set Y of its mixed strategies consists of all vectors y = (y 1, y 2,..., y n ) whose components satisfy y j = 1 and y j 0, j = 1,..., n.

16 Let us assume that players I and II are playing a zero-sum game determined by the matrix A. If I chooses the mixed strategy x, and II chooses y, then the expected payo is P(x, y) = a ij x i y j = x i a ij y j = y j a ij x i i.e. P(x, y) = x i P(i, y) = y j P(x, j). Here i in the expression P(i, y) denotes the i-th pure strategy of I. Analogously, j in the expression P(x, j) denotes the j-th pure strategy of II.

17 Lemma 3. Let (x 1, y 1 ) X Y (x 2, y 2 ) be saddle points of the payo-function P. Then (x 1, y 2 ) and (x 2, y 1 ) are also saddle points ofthe payo-function P. Proof. The denition of a saddle point implies that P(x 2, y 1 ) P(x 1, y 1 ) P(x 2, y 2 ) and P(x 1, y 2 ) P(x 2, y 2 ) P(x 2, y 1 ). These inequalities imply that P(x 2, y 1 ) = P(x 1, y 1 ) = P(x 2, y 2 ) = P(x 2, y 1 ).

18 Proof. (continuation) Let x and y be arbitrary mixed strategies from X and Y, respectively. Then we have that P(x, y 1 ) P(x 1, y 1 ) = P(x 2, y 1 ) = P(x 2, y 2 ) P(x 2, y), i.e. (x 1, y 2 ) is a also saddle point of the payo-function P. Analogously, one can prove that (x 2, y 1 ) is also a saddle point of the payo-function P. Lemma 4. Let ( x, ȳ) X Y be a saddle point of the payo-function P. If x i0 > 0, then P(i 0, ȳ) = P( x, ȳ). Also, if ȳ j0 > 0, then P( x, j 0 ) = P( x, ȳ).

19 Proof. Let us assume the contrary, i.e. P(i 0, ȳ) < P( x, ȳ). Since ( x, ȳ) X Y is a saddle point of the payo-function P, we have that P(i, ȳ) P( x, ȳ) for each i = 1, 2,..., m, with i i 0. Multiplying the both sides of these inequalities by x i, we obtain that x i P(i, ȳ) x i P( x, ȳ). After adding of all these m inequalities, we obtain that x i P(i, ȳ) < x i P( x, ȳ), i.e. P( x, ȳ) < P( x, ȳ).

20 The obtained contradiction shows that our assumption is wrong, and hence P(i 0, ȳ) = P( x, ȳ). Analogously, one can prove that P( x, j 0 ) = P( x, ȳ). This completes the proof. Remark. Let ( x, ȳ) X Y be a saddle point of the payo-function P. Then P( x, ȳ) is a value of the corresponding zero-sum game. Lemma 5. Let ( x, ȳ) X Y be a saddle point of the payo-function P determined by the matrix A = (a ij ) m n. Then ( x, ȳ) is a saddle point for the zero-sum game determined by the matrix B = (b ij ) m n with b ij = αa ij + β, where α > 0. Moreover, αp( x, ȳ) + β is the value of the zero-sum game determined by the matrix B.

21 Proof. Let us denote by P A and P B the payo-functions of the zero-sum games generated by the matrices A and B, respectively. Then P B (x, y) = b ij x i y j = (αa ij + β)x i y j = = α a ij + β x i y j = αp A (x, y) + β Because ( x, ȳ) X Y is a saddle point of the payo-function P A, the following inequalities hold true: for each x X and y Y. P A (x, ȳ) P A ( x, ȳ) P A ( x, y)

22 Proof of Lemma 5 (continuation). These inequalities imply that αp A (x, ȳ) + β αp A ( x, ȳ) + β αp A ( x, y) + β, i.e. P B (x, ȳ) P B ( x, ȳ) P B ( x, y) for each x X and y Y. Hence ( x, ȳ) is a saddle point of the payo-function P B and P B ( x, ȳ) is its value. Remark. Let A = (a ij ) m n be an arbitrary matrix and M := 1 + max a ij. We set B = (b ij ) m n with 1 i m,1 j n b ij = a ij + M, i = 1,..., m, j = 1,..., n. Then the all elements of the matrix B are positive, and hence, its value is also positive.

23 Lemma 6. Let ( x, ȳ) X Y be a saddle point of the payo-function P. If P(i 0, ȳ) < P( x, ȳ), then x i0 = 0. Also, if P( x, j 0 ) > P( x, ȳ), then ȳ j0 = 0. Proof. Let us assume that x i0 > 0. According to Lemma 4 we obtain that P(i 0, ȳ) = P( x, ȳ). This contradiction shows that x i0 = 0. Analogously, one can prove that ȳ j0 = 0. Proposition 1. Let S be a closed subset of R n and let x S. Then there exists point y S so that 0 < x y = min( x s : s S).

24 Proof of Proposition 1. Let d := inf( x s : s S) and let n be an arbitrary positive integer. Because d + 1 n > d there exists point y n S so that d x y n < d + 1 n. Because y n x y n + x d x, the sequence {y n } n=1 is bounded, and hence there exists a convergent subsequence {y nk } k=1 tending to y as k tends to innity. Taking a limit in the inequalities d x y nk < d + 1 n k as k, we obtain that d x y d. This completes the proof.

25 We denote by a, b := a i b i the scalar product of two vectors a = (a 1, a 2,..., a n ) and b = (b 1, b 2,..., b n ). Separability theorem: Let S be a closed convex subset of R n and let the point x does not belong to S. Then there exist a non-zero vector p and a real number q such that p, x = q and p, s > q for each s S. Proof. Let y S be a point of S such that x y = = min( x s : s S). We set p := y x and q := p, x. Then p, y q = p, y p, x = p, y x = p 2 > 0.

26 Proof (continuation). Let us assume that there exists a point s S so that p, s q. Then r := p, s y = p, s p, y < q q = 0. For each ε (0, 1) we set s ε := (1 ε)y + εs. The convexity of S implies that x ε S. On can directly checked that x x ε 2 = (x y) + ε(s y) 2 = x y 2 + 2ε y x, s y + ε 2 s y 2 = = x y 2 + ε ( 2r + ε s y 2) < x y 2 for each suciently small ε > 0 (because r < 0). The obtained contradiction completes the proof.

27 Theorem of the Alternative for matrices Let A = (a ij ) m n be an arbitrary real matrix. Then either (i) or (ii) must hold: (i) The point 0 R m belongs to the convex hull C of the vectors a 1, a 2,..., a n, e 1, e 2,..., e m, where a j := (a 1j, a 2j,..., a mj ), j = 1, 2,..., n, and e 1 = (1, 0,..., 0), e 2 = (0, 1, 0,..., 0),..., e m = (0, 0,..., 1). (ii) There exists a vector p = (p 1, p 2,..., p m ) such that p i = 1 and p i > 0 for each i = 1, 2,..., m, and p, a j > 0 for each j = 1, 2,..., n.

28 Proof. Let us assume that 0 C. Applying the Separability theorem, we obtain that there exist a non-zero vector p and a number q such that q = p, 0 = 0, p i = p, e i > 0 for each i = 1, 2,..., m, and p, a j > 0 for each j = 1, 2,..., n, i.e. (ii) holds true. The min-max theorem (von Neuman and Morgenstern) Let A = (a ij ) m n be an arbitrary real matrix. Then the zero-sum game determined by the matrix A has a value, i.e. vi A = vii A.

29 Proof of the min-max theorem. According to the Theorem of the Alternative for matrices, either (i) or (ii) must hold. If (i) is fullled, then there exists α 1 0, α 2 0,..., α n 0 and β 1 0, β 2 0,..., β m 0 such that α j + β i = 1 and α j a j + β i e i = 0,

30 Proof (continuation) i.e. α j a ij + β i = 0, i = 1, 2,..., m. If all α j = 0, j = 1, 2,..., n, then all β i = 0, i = 1, 2,..., m, and hence their sum can not be qual to 1. The obtained contradiction shows that at least one α j must be positive. We set ȳ j := α j n k=1 α k 0.

31 Proof (continuation) Clearly, each ȳ j, j = 1, 2,..., n, is well dened, ȳ j = 1 and α j a ij = β i 0, i = 1, 2,..., m, i.e. P A (i, ȳ) 0. This implies that for each mixed strategy x of the rst player P A (x, ȳ) = x i P A (i, ȳ) 0, and hence max x X PA (x, ȳ) 0. Thus we obtain that v II = min y Y max x X PA (x, y) 0.

32 Proof (continuation) Suppose, instead, that (ii) holds true, i.e. there exists a vector x = ( x 1, x 2,..., x m ) such that x i = 1 and x i > 0 for each i = 1, 2,..., m, and m x ia ij = x, a j > 0 for each j = 1, 2,..., n. Then x can be considered as a mixed strategy of the I player and P A ( x, j) > 0. This implies that for each mixed strategy y of the second player P A ( x, y) = y j P A ( x, j) > 0, and hence min y Y PA ( x, y) > 0. Thus we obtain that v I = max x X min y Y PA (x, y) > 0.

33 Proof (continuation) We know that v A I v A II. Let us assume that v A I < vii A. We consider + vii A. Then 2 the matrix B := (b ij ) m n with b ij := a ij v I A v B I v B II = v A I v I A = vii A v I A + vii A 2 + vii A 2 = v I A = v I A vii A 2 vii A 2 obtained contradiction shows that vi A proof of the Min-max theorem. < 0. Analogously, > 0. But this is impossible. The = v A II and completes the

34 Lemma 6. Let x and ȳ be mixed strategies of players I and II, and v be a real number such that P(i, ȳ) v P( x, j), i = 1, 2,..., m, j = 1, 2,..., n. Then v is the value of the game and the couple ( x, ȳ) - a Nash equilibrium. Proof. Let x and y are arbitrary mixed strategies of players I and II. Then one can check that P(x, ȳ) = x i P(i, ȳ) x i v = v = = y j v = y j P( x, j) = P( x, y).

35 Without loss of generality, we may think that the value v of a game determined by the matrix A = (a ij ) m n is positive (for, example, if all elements a ij > 0). Let v x and x ( v y and ȳ ) be solutions of the following linear problems v y min P(i, y) v y, i = 1, 2,..., m y j = 1 y j 0, j = 1, 2,..., n These problems can be written as follows v x max P(x, j) v x, j = 1, 2,..., n x i = 1 x i 0, i = 1, 2,..., m

36 v y min a ij y j v y, i = 1, 2,..., m y j = 1 y j 0, j = 1, 2,..., n v x max a ij x i v x, j = 1, 2,..., n x i = 1 x i 0, i = 1, 2,..., m These two linear problems are dual. If (v x, x) and (v y, ȳ) are the solutions of the rst and second problem, respectively, then v x = v y = v is the value of the game and the couple ( x, ȳ) is a Nash equilibrium.

37 If we set p i = x i /v, i = 1, 2,..., m, and q j = y j /v, j = 1, 2,..., n, then we can write the above problems as follows q j max p i min a ij q j 1, i = 1, 2,..., m a ij p i 1, j = 1, 2,..., n q j 0, j = 1, 2,..., n p i 0, i = 1, 2,..., m

38 Zero-sum games determined by a matrix of type 2 2 Let us consider a game determined by the following matrix ( ) a b c d Let us assume that this game has no Nash equilibrium in pure strategies. Let ( x, ȳ) be a Nash equilibrium with x = (1, 0) and ȳ = (ȳ 1, ȳ 2 ), where ȳ 1 > 0, ȳ 2 > 0 and ȳ 1 + ȳ 2 = 1. Then P(x, ȳ) P( x, ȳ) P( x, y). Clearly, v = P( x, ȳ) = aȳ 1 + bȳ 2. We set y = (1, 0) and y = (0, 1), and obtain that aȳ 1 + bȳ 2 min(a, b). This implies that a = b = v. Also, we have that v P(2, ȳ) = cȳ 1 + dȳ 2 min(c, d). If c = min(c, d), we obtain that ((1.0), (1, 0)) is a Nash equilibrium in pure strategies. If d = min(c, d), we obtain that ((1.0), (0, 1)) is a Nash equilibrium in pure strategies.

39 But, according to our assumption, the game has no Nash equilibrium in pure strategies. Hence the assumption that ( x, ȳ) be a Nash equilibrium with x = (1, 0) and ȳ = (ȳ 1, ȳ 2 ), where ȳ 1 > 0, ȳ 2 > 0 and ȳ 1 + ȳ 2 = 1, is not possible. Analogously it can be proved that it is not possible ( x, ȳ) to be a Nash equilibrium with x = (0, 1) and ȳ = (ȳ 1, ȳ 2 ), where ȳ 1 > 0, ȳ 2 > 0 and ȳ 1 + ȳ 2 = 1. This shows that whenever a zero-sum game has no a Nash equilibrium in pure strategies, then both components of the Nash equilibrium are mixed strategies.

40 Reducing zero-sum games Let us consider a zero-sum game determined by the matrix A = (a ij ) m n. Let the i 0 -pure strategy of I is dominated by a convex combination of the remainder pure strategies of the I player, i.e. a i0 j α i a i,j for each j = 1, 2,..., n,,i i 0 where all number α i are nonnegative and,i i 0 α i = 1.

41 We denote by A the matrix obtained from the matrix A be deleting the i 0 -raw. Let ( x, ȳ ) with x = ( x 1, x 2,..., x i 0 1, x i 0 +1,..., x m) be a Nash equilibrium for the game determined by the matrix A. Then ( x, ȳ ) is a Nash equilibrium for the game determined by the matrix A, where x = ( x 1, x 2,..., x i 0 1, 0, x i 0 +1,..., x m). Because ( x, ȳ ) is a Nash equilibrium for the game determined by the matrix A, the following inequalities hold true: P A (x, ȳ ) v = P A ( x, ȳ ) P A ( x, y ), for x X and y Y, where P A (x, y ) = a ij x i y j.,i i 0

42 In particular, we have that P A (i, ȳ ) v P A ( x, j), i = 1,..., i 0 1, i 0 +1,..., m, j = 1,..., n. i.e. a ij ȳ j v, i = 1,..., i 0 1, i 0 + 1,..., m,,i i 0 a ij x i v, j = 1,..., n (1) Since the vector α := (α 1, α 2,..., α i0 1, α i0 +1,..., α m ) can be considered as a mixed strategy of I, we have that P A (α, ȳ ) v,

43 i.e.,i i 0 a ij α i ȳ j v. From here (taking into account that the i 0 -pure strategy is dominated by a convex combination of the remainder pure strategies of the I player), it follows that a i0 jȳ j,i i 0 a ij α i ȳ j v. Applying this inequality and 1, we complete the proof taking into account Lemma 6.

44 Analogously, let us assume that the j 0 -pure strategy of II is dominated by a convex combination of the remainder pure strategies of the II player, i.e. a ij0,j j 0 β j a i,j for each i = 1, 2,..., m, where all number β j are nonnegative and,j j 0 β j = 1.

45 We denote by A the matrix obtained from the matrix A be deleting the j 0 -column. Let ( x, ȳ ) with ȳ = (ȳ 1, ȳ 2,..., ȳ j, ȳ 0 1 j,..., ȳ 0 +1 n) be a Nash equilibrium for the game determined by the matrix A. Then ( x, ȳ ) is a Nash equilibrium for the game determined by the matrix A, where ȳ = (ȳ 1, ȳ 2,..., ȳ j 0 1, 0, ȳ j 0 +1,..., ȳ n).

46 2 n and m 2 games We consider rst a zero-sum game determined by a the 2 n - matrix A. If v is its value, then v = max min x X y Y 2 a ij x i y j = min max y Y x X 2 a ij x i y j. Clearly, max x X min 2,2,...,n a ij x i v.

47 On the other hand 2 a ij x i y j = y j 2 a ij x i y j min,2,...,n 2 a ij x i = min,2,...,n 2 a ij x i, From here, it follows and hence min y Y 2 a ij x i y j min,2,...,n 2 a ij x i,

48 v = max min x X y Y 2 In such a way, we obtain that v = max x X a ij x i y j max x X min,2,...,n min,2,...,n 2 a ij x i. 2 a ij x i. Since X = {(x 1, x 2 ) : x 1 = α, x 2 = 1 α, α [0, 1]}, we obtain that v is the maximum (with respect to α) of the minimum of n linear functions (depending on α). So we can plot these function and to nd this maximum graphically.

49 2 n and m 2 games We consider next a zero-sum game determined by a the m 2 - matrix A. If v is its value, then v = min y Y max,2,...,m 2 a ij y j. Since Y {(y 1, y 2 ) : y 1 = β, y 2 = 1 β, β [0, 1]}, we obtain that v is the minimum (with respect to β) of the maximum of m linear functions (depending on β). So we can plot these function and to nd this minimum graphically.

50 Symmetric games A square matrix A = (a ij ) n n is said to be skew-symmetric if a ji = a ij for all i, j = 1,..., n. A zero-sum game is said to be symmetric if its matrix is skew-symmetric. Theorem. The value of a symmetric game is zero. If ( x, ȳ) is a Nash equilibrium, then ( x, x) and (ȳ, ȳ) are also Nash equilibriums. Proof. Because the matrix A is skew-symmetric, A T = A. Let x be an arbitray mixed strategy. Then x T Ax = (x T Ax) T = T A T x = = x T Ax, and hence x T Ax = 0.

51 Proof (continuation. Then 0 max x X min y Y x T Ax = v = min y Y max x X x T Ax 0 which implies that v = 0. Let j be an arbitrary index. Then P( x, j) 0, i.e. a ij x i 0, i.e. a ji x i 0, i.e. (2) a ij x j 0, i.e. P(i, x) 0. The last inequality, (2) and Lemma 6 complete the proof.

52 Theorem. A symmetric game has no a Nash equilibrium in pure strategies, if and only if one of the following assertions holds true: (i) each column has at least one positive element; (ii) each raw has at least one negative element. Proof. (suciency) We shall consider only (i). The case (ii) can be studied in the same way. Let (k, k) be a Nash equilibrium in pure strategies and let a ik > 0. Then 0 < a ik = P(i, k) v = 0. The obtained contradiction shows that there is no Nash equilibrium in pure strategies.

53 Proof. (necessity) Let us assume that there is no Nash equilibrium in pure strategies and there exists a column k such that a ik 0 for each i = 1, 2,..., n. This implies that a ik x i 0 for each mixed strategy x = (x 1, x 2,..., x n ), i.e. a ki x i 0, and hence a kj x j 0. So, we obtain that P(x, k) 0 and P(k, x) 0. This means that (k, k) is a Nash equilibrium in pure strategies.

54 Symmetric games Let us consider the matrix A = a 11 a a 1n a 21 a a 2n a m1 a m2... a mn and let us dene the following m + n + 1 m + n + 1 skew-symmetric matrix: B = O m m A m n I m 1 A T n m O n n I n 1 I1 m T I1 n T O 1 1 where O k k is a k k zero matrix and I1 m T components equal to 1. is a vector-raw with m

55 Theorem. Let the elements of the matrix A are positive. Let (( x, ȳ, λ), ( x, ȳ, λ)) be a Nash equilibrium of the game determined of the matrix B. Then (i) x i = ȳ j = µ > 0; (ii) the couple ( x/µ, ȳ/µ) is a Nash equilibrium of the game determined by the matrix A; (iii) the value of the game determined by the matrix A is λ/µ. Proof. We have that x i + ȳ j + λ = 1, with x i 0, i = 1,..., m, ȳ j 0, j = 1,..., n, and λ 0.

56 Proof (continuation). Since (( x, ȳ, λ), ( x, ȳ, λ)) is a Nash equilibrium of the game determined of the matrix B, the following inequalities hold true: a ij ȳ j λ 0, i = 1,..., m, (3) a ij x i + λ 0, j = 1,..., n, (4) x i ȳ j 0. (5)

57 Let us assume that λ = 0. This contradicts to (3). Let us assume that λ = 1. Then x = 0 and ȳ = 0. But this contradicts to (4). Hence λ (0, 1). Then x i + ȳ j = 1 λ =: 2µ and x i ȳ j = 0, i.e. x i = ȳ j = µ.

58 Then a ij ȳ j µ λ µ, x i µ a ij x i µ λ µ, ȳ i µ x i µ = ȳ j µ = 1 0, i = 1,..., m, 0, j = 1,..., n. Applying Lemma 6, we obtain that the couple ( x/µ, ȳ/µ) is a Nash equilibrium of the game determined by the matrix A.

59 Theorem. Let the elements of the matrix A are positive. Let the couple ( x, ȳ) be a Nash equilibrium of the game determined by the matrix A with value v. Then ( x 2 + v, ȳ 2 + v, v 2 + v ). is a Nash equilibrium of the game determined of the matrix B. Proof. Since ( x, ȳ) is a Nash equilibrium of the game determined by the matrix A, we have that x i = ȳ j = 1

60 Proof (continuation). a ij ȳ j v, x i 0, i = 1,..., m, a ij x i v, ȳ j 0, j = 1,..., n.

61 Proof (continuation). These inequalities imply that a ij a ij ȳ j v + 2 v v + 2 0, x i 0, i = 1,..., m, x i v v v + 2 0, ȳ j 0, j = 1,..., n. Applying Lemma 6, we obtain that the couple x i ( v + 2, ȳ j v + 2, v v + 2 ) is a Nash equilibrium of the game determined by the matrix B.