# ZERO-SUM TWO-PERSON GAMES

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1 Chapter 20 ZERO-SUM TWO-PERSON GAMES T.E.S. RAGHAVAN* University of Illinois at Chicago Contents References Appendix, by T. Radzik and T.E.S. Raghavan. Duels *The author would like to thank Ms Evangelista Fe, Swaminathan Sankaran and the anonymous referees for the many detailed comments that improved the presentation of this article. Handbook of Garne Theory, Volume 2, Edited by R.J. Aumann and S. Hart Elsevier Science B.V., 1994, All rights reserved

2 736 T.E.S. Raghavan Many parlor games like the game of "Le Her" or "Morra" [Dresher (1963)], involve just two players and finitely many moves for the players. When the number of actions available to a player is finite, we could in theory, reduce the problem to a game with exactly one move for each player, a move where actions for the players are various master plans for the entire game chosen from among the finitely many possible master plans. These master plans are called pure strategies. The original game is often analyzed using such a reduced form called games in normal form (also called strategic form). Certain information about the original game can be lost in the process [Kuhn (1953), Aumann and Maschler (1972)]. The reduction however helps to focus our understanding of the strategic behavior of intelligent players keen on achieving certain guaranteed goals against all odds. Given a two-person game with just one move for each player, the players independently and simultaneously select one among finitely many actions resulting in a payoff for each player. If i, j are their independent choices in a play, then the game is defined by a pair of real matrices A = (a~j), B = (bij) where aij is the payoff to player I and b~j is the payoff to player II. The game is called zero sum if aij + b~j = O. Thus in zero-sum games, what one player gains, the opponent loses. In such games A suffices to determine the payoff. We can use the following example [Dresher (1963)] to illustrate what we have said so far. Example. From a deck of three cards numbered 1, 2, 3 player I picks a card at will. Player II tries to guess the card. After each guess player I signals either High or Low or Correct, depending on the guess of the opponent. The game is over as soon as the card is correctly guessed by player II. Player II pays player I an amount equal to the number of trials he made. There are three pure strategies for player I. They are: ~: Choose 1, fl: Choose 2,?: Choose 3. For player II the following summarizes the possible pure strategies, excluding obviously "bad ones". (a) Guess 1 at first. If the opponent says Low, guess 2 in the next round. If the opponent still says Low, guess 3 in the next round, (b) Guess 1 at first. If the opponent says Low, guess 3 in the next round. If the opponent says High, guess 2 in the next round. (c) Guess 2 at first. If the opponent says Low, guess 3; if the opponent says High, guess 1. (d) Guess 3 at first. If the opponent says High, guess 1 in the next round. If the opponent says Low, guess 2 in the next round.

3 Ch. 20: Zero-Sum Two-Person Games 737 (e) Guess 3 at first. If the opponent says High, guess 2 in the next round. If the opponent still says High, guess 1 in the next round. Thus the payoff matrix is given by (a) (b) (c) (d) (e) A=fl A pair of pure strategies (i*,j*) for a payoff matrix A = (aij) is called a saddle point if a~j, ~ ai, j, ~< a~, i Vi, Vj. Thus a~,j, is the guaranteed gain for player I against any '~/" player II chooses. It is also the maximum loss to player II against any "i" player I chooses. In general such a saddle point may not exist. When it does we have ai,j, = max rain a.. = min max aij. i j 1J j i The above game has no saddle point. In fact max~ min t aij = 1 and min~ max~ a~j = 2. When players face intelligent opponents, the saddle point strategies are the best course of actions for both players. When a game has no saddle point, without violating the rules of the game, it might be possible for players to enlarge the available set of masterplans and look for a saddle point in the enlarged game. For example a master plan for a player in this new game could be based on the outcome of a suitably chosen random device from an available set of random devices, where the possible outcomes are themselves masterplans of the original game. If each player selects his masterplan based on the outcome of his independently chosen random device we enter into a new game called a mixed extension of the game. Here the set of available random devices are called mixed strategies for the original game and they will be the pure strategies for the mixed extension game. The expected payoff K(x, y) is the outcome for the mixed extension game when player I uses the random device x and player II uses the random device y. In our example above player I could rely on a random device x* -- (X*x, x2*, x~) = (_2! 2~ where say x* is the probability of choosing pure strategy ~ in the original 5 7 5~ 5/ game. Player I can guarantee an expected gain of 9 no matter which pure strategy * * * * * i 3 i player II uses. Similarly, the random device y* - (y,, Yb, Yc, Yd, Ye ) = (0, g, 3' g' O) where say y* = 3 is the probability of selecting the pure strategy c by the random device y*. Player II by using y* can guarantee an expected loss no more than 9 whatever player I does. Thus, if x, y are any arbitrary probability vectors for players I and II, then the expected payoff K(x, y) to player I satisfies K(x, y*) <~ K(x*, y*) <~ K(x*, y) Vx, and Vy.

4 738 T.E.S. Raghavan Thus for the mixed extension game, (x*, y*) is a saddle point. Here x*, y* are called optimal mixed strate#ies for players I and II respectively and 9 is called the value of the game. While Borel [von Neumann (1953)] suspected that games may not in general possess value even in the mixed extension, von Neumann (1928) Iaid the foundations of Game Theory by proving the decisive minimax theorem. This celebrated theorem remained dormant for a while. It was the classic monograph of [von Neumann and Morgenstern (1944)] that arrested the attention of mathematicians and social scientists and triggered research activity in Game theory. Of the many existence proofs for the minimax theorem [von Neumann (1928), Kakutani (1941), Ville (1938), Loomis (1946), Nash (1950), Owen (1970) and so on] Kakutani's proof and Nash's proof are amenable for extensions to more general situations. Nash's proof is based on the Brouwer's fixed point theorem. Brouwer's theorem asserts that any continuous self map q~ of a compact convex set X in Euclidean n-space R" admits a fixed point. That is q~(x)= x for some xex. We will now prove the minimax theorem using the Brouwer's fixed point theorem. Minimax theorem. (von Neumann). Let A = (ai~) be any m x n real matrix. Then these exists a pair of probability vectors x* =(xl,x, * *..,x*) and y* =(y*, y*... y*) such that for a unique constant v ~a~jx*>~v, j=1,2... n, (1) J a,jy* <~ v, i = 1, 2... m. (2) Equivalently if K (x, y) = ~,~ Zj ai jxiy j then (x*, y*) is a saddle point for K (x, y). That is min max K(x, y) = max min K(x, y), y x x y where min and max are taken respectively over the set of all probability vectors x for player I and probability vectors y for player II. Proof. Given any probability vector x for I and y for II ifpi = p~(x, y) = Zj aijyi - Zi Y'.j aijxlyj, qj = qj(x, y)= Zi Zj aijxiyj- Zi ai~xi, we are looking for a pair (x, y) with p~~<0 and qj~<0 for all i,j. Let q~ : (x, y) --* (, r/) be a map where ~ is a probability m-vector and t/is a probability n-vector defined by coordinates ~i xi + max(pi' O) yj + max(qj, O) =, t/j- ; i= 1,2... m, m n 1 + ~ max(p» 0) 1 + ~ max(q«, O) 1 1 j= 1,2,..,n. It can be easily checked that (p is a continuous self-map of the set of probability

5 Ch. 20: Zero-Sum Two-Person Games 739 vector pairs. By the Brouwer's fixed point theorem, there exists an (x*, y*) such that q~(x*, y*) = (x*, y*). Thus x* Sk max(pk, 0) = max (Pi, 0) for all i. For some i, x* > 0, Pi <~ O. Thus max(pk, 0) = 0 for all k. A similar argument shows that max(q~, 0) = 0 for all I. These are precisely the equalities we wanted for v = ~,~ Y'.j a~x*'iyj.* The second assertion that (x*, y*) is a saddle point for the mixed extension K(x, y) easily follows from multiplying respectively the inequalities (1) and (2) by yj and xi and taking the sum over j and i respectively. [] The existence theorem does not tell us how to compute a pair of optimal strategies. An algebraic proof was given by Weyl (1950). Thus if the entries of the payoff are rational, by the algebraic proof one can show that the value is rational and a pair of optimal strategies have rational entries. Brown and Von Neumann (1950) suggested a solution by differential equations that roughly mimics the above tatonnement process (x,y)~(~,tl). It turned out that the minimax theorem can be proved via linear programming in a constructive way leading to an efficient computational algorithm a la the simplex method [Dantzig (1951)]. Interestingly the minimax theorem can also be used to prove a version of the duality theorem of linear programming. Equivalence of the minimax theorem and the duality theorem Given an m x n real matrix A = (aii) and given column m-vector b and column n-vector c we have the following two problems called dual linear programs in standard form. Primal: max c.x subject to Ax <~ b, x >~ O, Dual: min b.y subjectto Ary >1 c, y >~0. Any x satisfying the constraints of the primal is called a feasible solution to the primal. Feasible solutions to the dual are similarly defined. Here c.x and b.y are called the objective functions for the primal and the dual. The following is a version of the fundamental theorem of linear programming due to yon Neumann [Dantzig (1951)]. Duality Theorem. If the primal and the dual problems have at least one feasible solution, then the two problems have optimal solutions; further, at any optimal solution, the value of the two objective functions coincide. Indeed the duality theorem and the minimax theorem are equivalent. We will see that either one implies the other.

6 740 T.E.S. Raghavan Duality theorem =~ minimax theorem Given a payoff matrix A = (ag j), by adding a constant c to all the entries, we can assume aij > 0 for all i,j and v > 0. Our inequalities (1) and (2) can be rewritten as aii v >~1' for all j, ~aij(yj~<~l, for all i, j kv/ where say, p = (Pl,..., P,ù), q = (ql... qù) with p,=xi~0, ~p~=l, (3) v i v qj=yj>~o, Zqj= 1-. (4) 1) j v Given a matrix A = a~j with all entries positive consider the dual linear programs Problem A. min ~i Pi subject to S~ a~jpi ~> 1 for all j and p~/> 0 for all i. Problem B. max Zj qj subject to Zj a,jqj ~< 1 for all i and qj ~> 0 for all j. Here q = 0 is feasible for Problem B. For large N the vector p = (N... N) is feasible for Problem A. By the duality theorem we have an optimal solution p*, q* to the two problems. Further 5Z~ p* = Zj q* at any optimal pairs. By normalizing p* and q* we have optimal x* and y* satisfying (1), (2) for the payoff A with value v = 1/Ziq*. Thus the minimax theorem is equivalent fo solving the above dual linear programs. Before reducing the dual linear programs to a single garne problem we need the following theorems. Theorem on skew symmetric payoffs. Let A = - A r be a payoff matrix. Then the value of the garne is zero and both players have the same set of optimal strategies. Proof. Let if possible v < 0. Let y be optimal for player II. Thus Ay = - AXy < 0 and hence ATy > 0. This contradicts v < 0. Similarly we can show that v > 0 is not possible. Thus v = 0 and optimal strategies of one player are also optimal for the other player. [] Equalizer theorem. Let A = (aij) be a payoff with value v. Let the expected payoff to player I when he uses any optimal strategy x and when player II uses a fixed

7 Ch. 20: Zero-Sum Two-Person Games 741 column be v. Then player II has an optimal strate#y which chooses this column with positive probability. Proof. We can as weil assume v = 0 and the particular column is the nth column. It is enough to show that the system of inequalities aiju j ~ 0 J u,= 1 has a solution u >/0. We could normalize this and get the desired strategy. This is equivalent to finding a solution u, w ~> 0 to the matrix equation [e A ólfu]=p=[~l (Here e n is nth unit vector). It is enough to show that the point p is in the closed cone K generated by the columns of the matrix Suppose not. Then by the strong separation theorem [Parthasarathy and Raghavan (1971)] we can find a strictly separating hyperplane (f, «) between the closed cone K and the point p. This shows that the vector f~> 0 and the scalar «< 0. Further f'aj i> 0 for all columns aj and f'an + «~> 0. Normalizing f we get an optimal strategy for player I which gives positive expectation when player II selects column n. This contradicts the assumption about column n. [] Minimax theorem=~duality theorem Consider the following m + n + 1 m + n + 1 skew symmetric payoff matrix B = - A T 0. b T _ c T Since B is a skew symmetric payoff the value of B is 0. Any optimal mixed strategy (t/*, ~*, 0) for player II is also optimal for player I. If 0 > 0, then the vectors y* = (1/0).11", x* = (1/0). 4" will be feasible for the two linear programming problems. Further when player Il uses the optimal strategy and I uses the last row, the expected income to player I is b.y* - c.x* <~ O. Since the inequality b.y* - c.x* >~ 0 is always true for any feasible solutions of the two linear programs, the two objective functions have the same value and it is easy to see that the problems have y*, x* as their optimal solutions. We need to show that for some optimal (t/*, ~*, 0),

8 742 T.E.S. Raghavan 0 > 0. Suppose not. Then by the equalizer theorem player II pays a penalty if he chooses column n with positive probability, namely -b. q* + c" ~* > 0. Thus either b.r/* < 0 or c-~*> 0. Say c. *> 0. For any feasible ~ for the primal and for the feasible y of the dual we have c.~ - b.y ~< 0. For large N, ~ = x + N~* is feasible for the primal and c.(x + N~*)-b'y > 0 a contradiction. A similar argument can be given when b.r/* < 0. Extreme optimal strategies. In general optimal strategies are not unique. Since the optimal strategies for a player are determined by linear inequalities the set of optimal strategies for each player is a closed bounded convex set. Further the sets have only finitely many extreme points and one can effectively enumerate them by the following characterization due to Shapley and Snow (t950). Theorem. Ler A be an m x n matrix garne with v v ~ O. Optimal mixed strategies x* for player I and y* for player II are extreme points of the convex set of optimal strategies for the two players if and only if there is a square submatrix B = (a~i)i~i,j~s such that i. B nonsin#ular. ii. ~i~~ ai~x* = v 111. ~ jej aijyj *~V iv. x* =0 ifi I. v. y* = 0 if jq~j. jöj ~ { 1, 2... n}. i~i«{1,2... m}. Proof. (Necessary.) After renumbering the columns and the rows, we can assume for an extreme optimal pair (x*, y*), x* > 0 i = 1, 2... p; y* > 0 j = 1, 2... q. If row i is actively used (i.e. x* > 0,) then Z~'=I ai;y* = v. Thus we can assume Ä = (aij)~~i,j~ä such that B aijx.{=v, jöj={1,2... q}, i=1 >v, forj~ß j=l aijy,{=v, i~t= {1,2...,p}, <v,, fori I. We claim that the p rows and the q columns of Ä = (ai;)ij,j~ j are independent. Suppose not. For some (rq... ~p) 0 i=1 p i~l aijni= O, je J, aijx* = v, jöj. Thus ~~'= 1 7~i Z) q'= 1 aiy* = v Z~=- ~. rt, = 0. As v ~ 0, Z pi=l zti = 0. Since Y~i aijx* > v for

9 Ch. 20: Zero-Sum Two-Person Garnes 743 jcjwe can find e > 0 sufficiently small such that x' i = x* - en i >~ 0, x"=x*+eni>~o, x'i = x'i' = ni - O, i > p and alix'i = ~ ai~x*-e ~, aijnl ~ v, for all j, i=1 i=1 i=1 a,jxt = ~, aijx* + e. ~ aijni >~ v, i=l i=1 i=1 for allj. Thus x', x" are optimal and x*= (x'+ x")/2 is not extreme optimal, a contradiction. Similarly the first q columns of Ä are independent. Hence a nonsingular submatrix B containing the first p rows and the first q columns of Ä = (a0id,jd and satisfying conditions (i)-(iv) exists. Conversely, given such a matrix B satisfying (i)-(iv), the strategy x* is extreme optimal, otherwise x' x, " x*= (x'+ x")/2 and we have Zi aiixi ' >~,i v, ~,~ a~jx~ "/> v for jöj with Y~i a~jx* = v for jöj. Thus Y.~~I a~~x'~ = ~.,iei %x'[ = v for j = J and the matrix B is singular. [] Since there are only finitely many square submatrices to a payoff matrix there could be only finitely many extreme points and as solutions of linear equations, they are in the same ordered subfield as the data field. The problem of efficiently loacting an extreme optimal strategy can be handled by solving the linear programming problem mentioned above. Among various algorithms to solve a linear programming problem, the simplex algorithm is practically the most efficient. Linear inequalities were first investigated by Fourier (1890) and remained dormant for more than half a century. Linear modeling of problems in industrial production and planning necessitated active research and the pioneering contributions of Kantorovich (1939), Koopmans (1951) and Dantzig (1951) brought them to the frontiers of modern applied mathematics. Simplex algorithm. max ~ bjyj J subject to Consider the canonical linear programming problem B aljyj=di, i=l,2... m, j=l yj ~>0, j=l,2,..,n. Any solution y = (Yl... yù) to the above system of inequalities is called a feasible solution. We could also write the system as yl C1 + y2 C2 + "'" -[- yn Cn : d, Yl,Y2,'",Yn ~ O,

10 744 T.E.S. Raghavan where C 1, C 2... C" are the columns of the matrix A and d is the column vector with coordinates d~, i = 1, 2... m. It is not hard to check that any extreme point Y = (Yl, Y2... yù) of the convex polyhedra of feasible solutions can be identified with a set of linearly independent columns C ~', C~~,..,C ~k such that the yj are zero for coordinates other than i 1, i 2... i k. By slightly perturbing the entries we could even assume that every extreme point of feasible solutions has exactly m coordinates positive. We call two extreme points adjacent if the line segment joining them is a one dimensional face of the feasible set. Algebraically, two adjacent extreme points can be identified with two bases which differ by exactly one basis vector. The new basis is chosen by bringing a column from outside into the current basis which in turn determines the removal of an appropriate column from the current basis. An iteration consists of searching for an improved value of the objective function at an adjacent extreme point. The algorithm terminates if no improvements in the value of the objective function with adjacent extreme points is possible. Fictitious play. An intuitively easy to implement algorithm to find the approximate value uses the notion of fictitious play [Brown (1951)]. Assume that a given matrix game, A = (a~~)m ù has been played for t rounds with (il,j1), (i»ja)..., (it,it) as the actual choices of rows and columns by the two players. Let rows 1, 2... m appear kl, ka... km times in the t rounds. For player II one way of learning from player I's actions is to pretend that the proportions (kl~t,..., km~t) are the true mixed strategy choices of player I. With such a belief, the best choice for player II in round t + 1 is to choose any column j~+ 1 which minimizes the fictitious expected payoff (1/t)~im=lk~a~j. Suppose in the above data, columns 1,2... n appear l~,l 2... lù times in the first t rounds. Player I can also pretend that the true strategy of player II is (Ix/t, 12/t... lù/t). With such a belief the best choice for player I in round t + 1 is to choose any row t t + 1 which maximizes the fictitious expected income 1 /t Z~ n = ~ lja~j. The remarkable fact is that this naive procedure can be used to approximate the value of the garne. We have the following Theorem. Let (x t, y') be the strategies (kl~t,..., km~t), (Il~t,...,lù/t) t = 1, 2... where (xl,y ~) is arbitrary and (xt, J)for t>~2 is determined by the above fictitious play. Then v = lim min 1 kiai~ = lim max ~ ~ ljai~. t-~ o9 j t i = l t--* ao i [j=l The above procedure is only of theoretical interest. It is impractical and the convergence to the value is known to be very slow. Even though v(t)= min,(i/t) y,m 1 kiaij--* V as t ~ o% the mixed strategies (t) = (kl~t, k2/t,..., k~/t) and t/(t) = (l~/t..., IJt) may not converge.

11 Ch. 20: Zero-Sum Two-Person Garnes 745 A proof can be given [Robinson (1951)] by showing that for any skew symmetric payoff A, lim m!n ~ klalj = O. t.t i In general the sequence of strategies { (x t, yt)} o scillates around optimal strategies. Completely mixed games. A mixed strategy x for player I is called completely mixed iff x > 0 (i.e. all rows are essentially used). Suppose x, y are completely mixed optimal strategies for the two players. The inequalities Ay ~< v" 1 are actually equalities, for otherwise v = (x, Ay)< (x,v.1)= v, a contradiction. In case v = 0, the matrix A is singular. We call a matrix garne A completely mixed if every optimal strategy is completely mixef for both players. Theorem. If a matrix game A is compietely mixed, then A is a square matrix and the optimal strategies are unique. Proof. Without loss of generality v 0. In case y' y" are two extreme optimal strategies for player II, then Ay'= v. 1, Ay"= v. 1 and A(y'-y")= 0. Thus rank A < n. Since the extreme y' > 0, by the Shapley Snow theorem, A has an n x n submatrix which is nonsingular. This contradicts rank A < n. Thus rank A = n and the extreme optimal strategy is unique for player II. A similar argument applies for player I and shows that rank A =m, and the extreme optimal strategy is unique for player I. [] We have a formula to compute the value v for completely mixed garnes, and it is given by solving Ay = v.1. The unique solution y is optimal for player II. Since y is a probability vector y = va gives v =det A/(Zl Z~ Aij) where det A is the determinant of A and Aij are the cofactors of A. In case the payoff is a square matrix it can be shown that when one player has on optimal strategy which is not completely mixed then his opponent also possesses an optimal strategy that is not completely mixed [Kaplansky (1945)]. For Z-matrices (squate matrices with oft-diagonal entries nonpositive) if the value is positive, then the maximizer cannot omit any row (this results in a submatrix with a nonpositive column which the minimizer will choose even in the original garne). Thus the game is completely mixed. One can infer many properties of such matrices by noting the game is completely mixed. It is easy to check that since v > 0 the matrix is non-singular and its inverse is nonnegative: For completely mixed garnes A = (aij) with value zero, the cofactor matrix (Aij) has all Aij > 0 or all A~j < 0. This can be utilized to show that any Z-matrix with positive value has all principal minors positive [Raghavan (1978), (1979)]. The reduction of matrix garnes to linear programming is possible even when the strategy spaces are restricted to certain polyhedral subsets. This is useful in

12 746 T.E.S. Raghavan solving for stationary optimal strategies in some dynamic garnes [Raghavan and Filar (1991)]. Polyhedral eonstraints and matrix games. Consider a matrix game A where players I(II) can use only mixed strategies constrained to lie in polyhedra X(Y). Say X = {x:b'rx <~ c, x >>. O} where X «set of mixed strategies for player I. Let Y={y:Ey~>f y~>o}. We know that max~~ x minr~ r X T Ay = miny~r maxx x X T Ay. The linear program minr~ r x T Ay has a dual maxf. z, zet where T= {z:etz <~ ATx, z >1 0}. Thus player I's problem is max f.z, (z,x)ek where K = {(z, x): ETT, <~ X, BTx <~ c, z >~ O, X >~ 0}. This is easily solved by the simplex algorithm. Dimension relations. Given a matrix game A = (aij)m ù let X, Y be the convex sets of optimal strategies for players I and Il. Ler the value v = 0. Let Ja = {J:Yj > 0 for some y~y}. Let Jz = {J:Z~a~ix~=O for any x~x}. It is easy to show that Jl c Jz. From the equalizer theorem we proved earlier that J2 c J1. Thus Ja = J2 = J, say. Similarly we have 11 = 12 = I for player I. Since the rows outside 12 and the columns outside JE are never used in optimal plays, we can as well restrict to the garne Ä with rows in I2 and columns in Jz and with value 0. Let )~, Y be the vector spaces generated by X, Y, respectively. The following theorem independently due to [Bohnenblust, Karlin and Shapley (1950)] and [Gale and Sherman (1950)] charaeterizes the intrinsie dimension relation between optimal strategy sets and essential strategies of the two players. Dimension theorem. Il] - dim )~ = I JI - dim Y. Proof. Consider the submatrix Ä = (alj)i~t.»«for any y~ Y let )7 be the restriction of y to the coordinates j~j. We have Ay = 0. Ler An = 0 for some n in R I 1. Since we can always find an optimal strategy y* with y* > 0 for all j~j, we have z = y* - en >~ 0 for small e and Äz = 0. Clearly z~ Y. Further since the linear span of y* and z yield n the vector space {u:äu=0} coincides with Y. Thus dim Y= I J[- rank Ä. A Similar argument shows that dim )~ = III- tank Ä. Hence rank Ä = I II-dim 7~= I JI-dim Y. []

13 Ch. 20: Zero-Sum Two-Person Garnes 747 Semi-infinite games and intersection theorems Since a matrix payoff A = (afj) can be thought of as a function on I x J, where iei= {1,2,..,m} and jsj= {1,2...,n}, a straighfforward extension is to prove the existence of value when I or J is not finite. If exactly one of the two sets I or J is assumed infinite, the games are called semi-infinite garnes [Tijs (1974)]. Among them the so-called S-games of Blackwell and Girshick (1954) are relevant in statistical decision theory when one assumes the set of states to be finite of nature. Let Y be an arbitrary set of pure strategies for player II. Let I = { 1, 2..., m} be the set of pure strategies for player I. The bounded kernel K(i, y) is the payoff to player I when "i" is player I's choice and ye Y is the choice of player II. Let S = {(S1, \$2,.., Sm)IS i = K(i, y), i = 1, 2,.., m; y~ Y}. The garne can also be played as follows. Player II selects an ses. Simultaneously player I selects a coordinate i. The outcome is the payoff sf to player I by player II. Theorem. If S is any bounded set then the S-garne has a value and player I has an optimal mixed strategy. If con S (convex hull of S) is closed, player II has an optimal mixed strategy which is a mixture of at most m pure strategies. If S is closed convex, then player II has an optimal pure strategy. Proof. Let t* ~ T = con S be such that mint«max~ tl = maxi t* = v. Let (4, x) = c be a separating hyperplane between Tand the open box G = {x:maxi xl < v}. For any e > 0 and for any i, t* - e < v. Thus ~~ ~> 0 and we can as weil assume that is a mixed strategy for player I. By the Caratheodory theorem [Parthasarathy and Raghavan (1971)] the boundary point t* is a convex combination of at most m points of S. It is easy to check that c = v and ~ is optimal for player I. When S is closed the convex combination used in representing t* is optimal for player II; otherwise t* is approximated by t in S which is in the e neighborhood of t*. [] The sharper assertions are possible because the set S is a subset of R m. Many intersection theorems are direct consequences of this theorem [Raghavan (1973)]. We will prove Berge's intersection theorem and Helly's theorem which are needed in the sequel. We will also stare a geometric theorem on spheres that follows from the above theorem. Berge's interseetion theorem. Let S 1, \$2,.., Sc be compact convex sets in R m. Let (-I,~jSi#Oforj= 1,2... m. IfS= Uk=~ Sf is convex then ~k=~ Sf # ~b. Proof. Let players I and II play the S-game where I chooses one of the indices i= 1, 2,.., k and II chooses an x~s. Let the payoff be fr(x)= distance between x and Si. The functionsfi are continuous convex and nonnegative. For any optimal

14 748 T.E.S. Raohavan #flx~ + P2lx~ + "'" + ltpl«~ of player II, we have Thus player II has an optimal pure strategy x = Zj#ix» If an optimal strategy 2 = ()~,... 2«) of player I skips an index say 1, then player II can always choose any Y~Ni~ i S~ and then 0 = Zi 2if~(y) ~> v. Thus v = 0 and x o e(-]i= k 1 S~. If21>0Vi, then fi(x ) = v. Since x esi, for some i we have v = 0 and x e N~= i Si. [] Helly's theorem. k m+l. Let S1,S2,., Sk be convex sets in R m. Let NiEISi ~ ~b/fl ~ Ill Then Ni=xSi~~. Proof. By induction we can assume that for any II[=r~(m+l), ~i«xs~:/:(o and prove it for I/I--r+ 1. Say, I= {1,2... r+ 1}. Let aiss ~ if 14:j. Let C= con{a~... at~+x)}. Since r>n by the Caratheodary theorem C= [..)i~~ Ci where Ci = ton {ai... ai- i, ai+ l, aù+ i}. (Here we define a o = aù+ i and aù+ 2 = ai). Further C i ~ S i. By Berge's theorem ~i~t Ci :/: c~. [] The following geometric theorem also follows from the above arguments. Theorem. Let S1,\$2»..., S m be compact convex sets in a Hilbert space. Let (-]i~ j Si v ~ qb for j = 1, 2... m, but 0 m= i i Si = q~. Then there exists a unique v > 0 and a point x o such that the closed sphere S(xo, v) with center x o and radius v has nonnull intersection with each set S i while spheres with center Xo and with radius < vare disjoint with at least one S i. In fact no sphere of radius < v around any other point in the space has nonempty intersection with all the sets S i. When both pure strategy spaces are infinite, the existence of value in mixed strategies fails to hold even for very simple garnes. For example, if X -- Y = the set of positive integers and K(x,y)= sgn(x- y), then no mixed strategy p = (Pl, P2... ) on X can hedge against all possible y in guaranteeing an expected income other then the worst income - 1. In a sense if p is revealed, player II can select a sufficiently large number y such that the chance that a number larger than y is chosen according to the mixed strategy p is negligible. Thus supinfk*(p,y)= -- 1 where K*(p,y)=~p(x)K(x,y). P Y x A similar argument with the obvious definition of K*(p, q) shows that sup infk*(p, q) = -- 1 < infsup K*(p, q) = 1. P q q q The failure sterns partly from the noncompactness of the space P of probability measures on X.

15 Ch. 20: Zero-Sum Two-Person Garnes 749 Fixed point theorems for set valued maps With the intention of simplifying the original proof of von Neumann, Kakutani (1941) extended the classical Brouwer's theorem to set valued maps and derived the minimax theorem as an easy corollary. Over the years this extension of Kakutani and its generalization-i-glicksberg (1952)] to more general spaces have found many applications in the mathematical economics literature. While Brouwer's theorem is marginally easier to prove, and is at the center of differential and algebraic topology, the set valued maps that are more natural objects in many applications are somewhat alien to mainstream topologists. Defnition. For any set Y let 2 r denote a collection of nonempty subsets of Y. Any function ~b:x~2 r is caued a correspondence from X to Y. When X, Y are topological spaces, the correspondence ~b is upper hemicontinuous at x iff given an open set G in Yand given G = ~b(x), there exists a neighborhood N~x, such that the set q~(n) = Ur~s q~(y) c G. The correspondence q~ is upper hemicontinuous on X iff it is upper hemicontinuous at all xex. Kakutani's fixed point theorem. Let X be compact convex in R". Ler 2 x be the collection of nonempty compact convex subsets of X. Let c~:x ~ 2 x be an upper hemi continuous correspondence from X to X. Then x~q~(x) for some x. In order to prove minimax theorems in greater generality, Kakutani's theorem was further extended to arbitrary locally convex topological vector spaces. These are real vector spaces with a Hausdorff topology admitting convex bases, where vector operations of addition and scalar multiplication are continuous. The following theorem generalizes Kakutani's theorem to locally convex topological vector spaces [Fan (1952), Glicksberg (1952)]. Fan-Glicksberg fixed point theorem. Let X be compact convex in a real locally convex topological vector space E. Let Y be the collection of nonempty compact convex subsets of X. Let (p be an upper hemicontinuous correspondence from X to YI Then x~(9(x) for some x. The following minimax theorems of [Ville (1938) and Ky Fan (1952)] are easy corollaries of the above fixed point theorem. Theorem. (Ville). Ler X, Y be compact metrtc spaces. Let K(x, y) be continuous on X x Y Then minmaxffk(x,y)du(x)dv(y)=maxminffk(x,y)dv(x)d~(y)v» ~ v where I~, v may range over all probability measures on X, Y, respectively.

16 750 T.E.S. Raohavan Ky Fan's minimax theorem. Let X, Y be compact convex subsets of locally convex topological vector spaces. Ler K:X x Y-~ R be continuous. For every 2EX, ~~ Y, let K(~, y): Y-~ R be a convex function and K (x, )~): X ~ R be a concave function. Then Proof. and max min K(x, y) = min max K(x, y). xex y~y yey xex Given any 2~X, y~ylet A(~) = {v:ve Y, mink()~,y) = K(~,v)} yey F(y) = {u:u X, mink(x,y) = K(u,)7)}. xex The sets A(~) and F07 ) are compact convex and the function q~:(x, y)~ F05 ) x A(2) is an upper hemicontinuous map from X x Y to all nonempty compact convex subsets of X x Y. Applying Fan-Glicksberg fixed point theorem we have an (x,y )eó(x,y ). This shows that (x,y ) is a saddle point for K(x,y). [] In Ky Fan's minimax theorem, the condition that the function K is jointly continuous on X x Y is somewhat stringent. For example let X = Y = the unit sphere S of the Hilbert space 12 endowed with the weak topology. Let K(x, y) be simply the inner product (x, y). Then the point to set maps y ~ A(y), x ~ F(x) are not upper hemicontinuous. [-Nikaido (1954)]. Hence Ky Fan's theorem is inapplicable to this case. Yet (0, 0) is a saddle pointi However in Ville's theorem, the function K*(l~,v)=~~K(x,y)d#(x)dv(y) is bilinear on P x Q where P(Q) are the set of probability measures on X(Y). Further K* isjointly continuous on P x Q where P, Q are compact convex metrizable spaces viewed as subsets of C*(X)(C*(Y)) in their weak topologies. Ville's theorem now follows from Fan's theorem. Definition. Let X be a topological space. A function f:x ~ R is called upper semicontinuous iff {x:f(x) < c} is open for each c. If g is upper semicontinuous, -g is called lower semicontinuous. In terms of mixed extensions of K on X Y, the following is a strengthening of Ville's theorem. Theorem. Let X, Y be compact Hausdorff. Let K:X Y~R be such that K(x,-) and K(',y) are upper semicontinuous and bounded above for each xöx, ye Y. Then inf sup K*(#, v) = sup inf K*(#, v), veq,u~p p_~p wq where P(Q) are the set of probability measures with finite support on X(Y).

17 Ch. 20: Zero-Sum Two-Person Garnes 751 Proof. Let X*, Y* denote regular Borel probability measures on X and Y respectively. When X or Y is finite the assertion follows from Blackwell's assertion for S-games. Let Y be any finite subset of Y. For e > 0, the set of e-optimal strategies # of the maximizer in the mixed extension K* on X* ~'* is weakly compact. This decreasing net of e-optimals have a nonempty intersection with inf max K*(#, v) = max inf K*(#, v). (7) veq /~ex*,uex* veq For Ymetrizable, K* is well-defined on X* Y*. However by (7) inf max K*(#, v) ~> inf sup K*(/A v) veq,uex* veq #ep max inf K*(#, v) ~< sup inf K*(#, v). #ex* veq #ep veq Thus the assertion follows when Y is metrizable. The general case can be handled as follows. Associate a family G of continuous functions ~b on Ywith K(x, y) >~ ~(y) for some x. We can assume G to be countable with infsup I49(y)dv(y)= inf fsup K(#, v). veq ~beg J veq J/~eP Essentially G can be used to view Yas a metrizable case. [] Other extensions to non-hausdorff spaces are also possible [Mertens (1986)]. One can effectively use the intersection theorems on convex sets to prove more general minimax theorems. General minimax theorems are concerned with the following problem: Given two arbitrary sets X, Yand a real function K:X Y-~R, under what conditions on K,X, Y can one assert sup inf K(x, y) = inf sup K(x, y). xex yey yey xex A standard technique in proving general minimax theorems is to approximate the problem by the minimax theorem for matrix games. Such a reduction is often possible with some form of compactness of the space X or Y and a suitable continuity and convexity or quasi-convexity on the function K. Definition. Let X be a convex subset of a topological vector space. Let f:x ~ R. For convex functions, {x:f(x)< c} is convex for each c. Generalizing convex functions, a functionf:x ~ R is called quasi-convex if for each real c, {x:f(x)< c} is convex. A function g is quasi-concave if -g is quasi-convex. Theorem. [Sion (1958)]. Ler X, Y be convex subsets of linear topological spaces with X compact. Ler K:X x Y~R be upper semicontinuous in x (for each fixed y) and lower semicontinuous in y (for each x). Let K(x,y) be quasi-concave in x and

18 752 T.E.S. Raghavan quasi-convex in y. Then sup inf K(x, y) = inf sup K(x, y). x~x y =Y x~y x~x Proof. Case (i). Both X, Yare compact and convex: Let ifpossible supx infy K(x, y) < c < infrsup~k(x,y ). Let A z = {y:k(x,y) > c} and B r = {x:k(x,y) < c}. Therefore we have finite subsets A c X, B «Y such that for each ye Y and hence for each y~con B, there is an x~a with K(x,y)> c and for each x~x and hence for each xecon A, there is a y~b, with K(x,y)< c. Without loss of generality let A,B be with minimal cardinality satisfying the above conditions. We claim that there exists an xoecona such that K(xo, y)< c for all y~b and hence for all yecon B [by qu.asi-convexity of K(xo, y)]. Suppose not. Then if B= {Yo, Yl...,yù}, for any -~ x~x there exists a yjeb such that K(x, yj)>~c. Let Ci=(x:K(x, yi)>~c}. The minimality of B means that (-]7=o.i~j Ci ~ ob. Further (~7=o C~ = q~. By Helly's theorem the dimension ofcon B is n. Thus it is an n-simplex. IfG~ is the complement of Cg, then the G~ are open and since every open set G~ is an F«set the G:s contain closed sets H~ that satisfy the conditions of Kuratowsky-Knaster-Mazurkievicz theorem [Parthasarathy and Raghavan (1971)]. Thus we have NT=o Hi ~ (9, and that ~7=o Gg ~b. That is, for some xo~cona, K(xo, y)<c for all y. Similarly there is a yoecon B, such that K(x, Yo) > c for all x~con A. Hence c < K(xo, Yo) < c, a contradiction. Case (ii). Let X be compact convex: Let supx infy K < c < infy SUpx K. There exists B c Y, B finite, such that for any xex, there is a yeb with K(x,y)<c. The contradiction can be established for K on X x Con B c X x Y. [] Often, using the payoff, a topology can be defined on the pure strategy sets, whose properties guarantee a saddle point. Wald first initiated this approach [Wald (1950)]. Given arbitrary sets X, Y and given a bounded payoff K on X x Y, we can topologize the spaces X, Ywith topologies :-x, 3--r where a base for ~-x consists of sets of the type S(xo, e ) = [x:k(x, y)- K(xo, y ) < e for all y}, e > 0, xo~x. Definition. The space X is conditionally compact in the topology Æx iff for any given e > 0, there exists a finite set {x 1, x 2... xù~~)} such that U~'~)~ S(xi, e) = X. The following is a sample theorem in the spirit of Wald. Theorem. Let K:X Y-~R. Ler X be conditionally compact in the topology 9- x. For any 6>0 and finite sets A~X, B«Y let there exist Y~X,y~Y such that K(x, y) <~ K(~, y) + 6for all x~a, y~b. Then Supx infr K(x, y ) = infr supx K(x, y ).

19 Ch. 20: Zero-Sum Two-Person Games 753 Proof. Since X is conditionally compact in the topology ~, for e > 0 inf sup K ~< infmax K + e Y X Y A for a finite set A ~ X. If B is any finite subset of Y, then by assumption inf r max a K ~< SUPx min» K. Thus inf sup K ~< infsup min K + e, Y X X B where ~ is the collection of finite subsets of Y. Since X is J-x conditionally compact, the right side of the above inequality is finite or - ~. We are through if v = infsup min K ~< sup infk + 2e. For otherwise ifsupinfk<v-2e, wehave x B x r X Y infk(x, y) < v - 2e, r and thus for any finite set A infk(x~,y) < v- 2e, r for all x for all x~~a. That is for each x i, and e > 0, there exists a y~ such that K(x~, y~) <~ K(x~, Yi) + e <~ v -- e for each i. Thus supmink(x,y~) <~ v - e, X B That is v ~< v - e, a contradiction. [] where B = {y~, iel with 1I[ finite}. Continuous payoffs. While the above general minimax theorems guarantee e- optimals with finite steps, in mixed extensions, certain subclasses of games admit optimal mixed strategies with finite steps. These subclasses depend on the nature of the cones generated by K(x,.) and K(.,y). Let X, Y be compact metric. Let K: X x Y~ R be continuous. Since K is bounded, we can assume K > 0. Let C(X), C(Y) be the Branch space of continuous functions on X, Y, respectively. The functions h«(x)= K(x,«) generate a cone whose closure we denote by C. Let E = C - C. Let P be the cone of nonnegative functions in C(X). Any positive linear operator A:E ~ C(X) maps C into P. Theorem. Let C have non-null interior in Ë or let A be an isometry and the image cone A(C) have non-null relative interior in its closed linear span. Then player II (the minimizer) has an optimal strategy with finite spectrum. If C has nonnull interior, both players have finite step optimals. Further, in this case K(x, y) is separable. That is K(x, y) = Zi ~j ai~ri(x)sj(y). Proof. For any Borel probability measure v on Y the map z:v~k*(v) is a continuous map of Y* into C(X). Further, B = z(y*) is compact in C(X). In fact

20 754 T.E.S. Raghavan when A is an isometry, S = con (A(B), 0) is compact and the range cone A(C)= Uù~= 1 ns. By the Baire Category Theorem the compact set S has nonnull interior and hence finite dimensional. The isometry A preserves extreme points. The function K(., et) is mapped into M(', «) under the isometry and any finite step optimal Vo for M(x, y) is optimal for K. Separability of K follows from the finite dimensional nature of C when C has interior. [] Extreme optimals. Extreme optimal strategies are hard to characterize for infinite games except for some special cases. Let X, Y be compact metric with a continuous payoff K(x,y). Then the following theorem extends Shapley-Snow theorem to certain infinite games. Theorem. Let v # 0 and ler v o be optimal for player II with spectrum a(vo)= Y Then an optimal strategy # with spectrum tr(l~ ) = X o is extreme iff {h«(x):k(x, et), cte Y} generates a dense linear manifold in LI (Xo, ~, Iz), where ~ is the class of Borel sets on X o. The proof mimics Shapley-Snow arguments for finite games with the helpful hint that L ~ = L*. Needless to say, the structure of optimal strategies can be quite complicated even for C ~ payoffs K on the unit square. An example will settle what we want to convey l-glicksberg and Gross (1953)]. Example. Let #~, vn be the nth moments of any arbitrary probability measures # # v with infinite spectrum. Then the C ~ kernel K (x,y) = Z~=o(1/2~)(x ~o ~- #ù) (f-v~) has/~, v as the unique optimal strategies. Proof. The kernel K is analytic and for any optimal 2 for player I we have Zù~o(1/2n)(2n-/~ù)(yn-v~)=0. Thus 2ù =/~~ for all n=~2=#. Similarly v is the unique optimal for the minimizer. [] More generally, given compact convex sets S, Tof probability measures on the unit interval, one would like to know when they would be the precise set of optimal mixed strategies for a continuous game on the unit square. The following theorem is a partial answer to this question [Chin, Parthasarathy and Raghavan (1976)]. Theorem. Ler S, T be compact convex sets of probability measures on the unit interval with only finitely many extreme points given by {#1, ]/2... /A p } and {v 1, v 2... v q} respectively. Let the spectrum of at least one #es, v~ T be the entire unit interval. Further for any e > 0, let Bi(E j) ùt max sup max sup t~.(hj) < 1 <j<q ißj ~ < F., 1 <_j<_q icj # o,j (Hj) 6.

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