Games of economic survival

zyxwv zyxwv zyxwvuts GAMES O F ECONOMIC SURVIVAL* M a r t i n Shubik a n d G e r a l d L. Thompson t G e n e r a l E l e c t r i c Go. and O h i o W e s l e y a n U n i v e r s i t y zy 1. INTRODUCTION The games examined in this paper were originally formulated by one of the authors in order to investigate some problems in economic theory pertaining to the theory of the firm [4]. A corporation has to give consideration both to its prospects for survival and to its ability to pay dividends. Survival p e r s e could be a goal, as could the maximization of the discounted value of the dividend payments. The latter might even involve the eventual ruin of the firm as part of the optimal policy. In order to portray the relationship between the market and the financial aspects of the firm, we can construct a type of game whose form i s closely related to the Gambler's Ruin problems. The fortunes of the firm a r e divided into a corporate and a withdrawal account. Ruin conditions apply to the former, and dividend payments a r e made in the latter. In general, complex dynamic problems a r e soon encountered. This paper deals only with simple examples involving one and two person dynamic nonzero sum games. zyxw 2. ONE-PERSON GAMES O F ECONOMIC SURVIVAL 2.1. General Description of One-Person Games of Economic Survival A one-person game of economic survival is characterized by the following quantities: a discrete set of times t = 0, 1, 2, ; a set of integers ai, i = 1, , n that represent the amounts that a player obtains each time he plays; a set of positive numbers pi, i = 1, , n, with C pi = 1, where pi represents the player's probability of obtaining ai when he plays and where successive plays of the game a r e assumed to be independent; a corporate account whose size at time t is denoted by C(t); a withdrawal account whose size at time t is denoted by W(t); a discount r a t e p (< 1) which is effective on the withdrawal account only; a ruin payment F that is paid to the player upon ruin or on leaving the game by withdrawing all assets; a ruin level B and a ruin condition that says, if C (t - 1) > B and C (t) 5 B the player received the ruin payment F and the game is terminated at time t. At the beginning of any time period he may transfer any integral p a r t of his corporate account into his withdrawal account; let u s indicate such a transfer by w. Such a payment may be interpreted a s a dividend payment to stockholders and is nonrecoverable, hence payments . .. ... ... zyx *Manuscript r e c e i v e d September 1 2 , 1958. t T h e a u t h o r s wish t o e x p r e s s t h e i r gratitude to D r s . Lloyd Shapley a n d Alan Hoffman f o r valuable c r i t i c i s m . Additionally, t h e a u t h o r s a r e indebted t o M r . R . Singleton f o r s e v e r a l i n t e r e s t i n g c o m m e n t s and o b s e r v a t i o n s , including an a l t e r n a t i v e proof f o r t h e solution to the o n e - p e r s o n game. 111 112 zyxwvutsrqpo zyxwvutsr zyxwvutsrq M. SHUBIK AND G. L. THOMPSON can never be made from the withdrawal to the corporate account. Let the initial amount in the corporate account be C ( 0 ) = x (an integer). The goal of the player is to maximize the discounted expected value of his withdrawal account. This completes the description of the game. Suppose for a moment there were no transfer privileges. Then if we watch the corporate account it varies as if it were a random walk on the line. Thus it moves from a value C (t) at time t to the value C (t + 1) = C (t) + ai at time t + 1 with probability pi. If there a r e transfer privileges, then, since we have assumed that each time the game i s played the probabilities of obtaining payments ai are independent of previous plays of the game, we see that the transfer payment w should be a function only of the value of the corporate account. Therefore, i f transfer payments w (C (t)) a r e included, the corporate account moves from C (t) to C (t + 1) = C (t) + ai - w (C (t)) with probability pi for i = 1, * * ' , n. A financial strategy (or simply, strategy) is a transfer payment function w (x). (The function w is from positive to non-negative integers.) Let W be the set of all financial strategies. Given a strategy w, one can compute the value function V (x, w) that measures the worth to the player of playing the game with initial a s s e t s x and strategy w. The value function gives the expected discounted value of the player's withdrawal account. In order to compute the expected discounted value of the withdrawal account, given initial assets x and withdrawal strategy w, we consider truncated games of length n, that is, games which a r e played at times t = 0, 1, - ' * , n. Let sn be the outcome of a series of plays of the subgames played each period. This indicates whether the player has obtained the maximum o r minimum revenue from the first play of the game, the second, etc., up to the nth play. From this information we can compute the amount K (sn,w, t) that is added to the player's witndrawal account for each of the times t = 0, 1, * , n. Let q (sn)be the probability that sn occurs. Then, taking sums and passing to the limit, we have as the expected discounted value of the withdrawal account. In order to show that this limit exists, we consider the most favorable game-the one in which the player wins every time he * plays. For this game, the most favorable strategy is w , for which the player withdraws all his assets above the ruin level as soon as he can. Then zyxwvuts zyxwvutsr * V ( X , W ) = ( x - B) + 2 pt = ( x - B) t=1 +- P 1-P zyxw zyxw for this game. And, obviously, for any other course of the game and any other strategy w, we * have 0 5 V (x, w) 5 V (x, w ). Since the expression for V (x, w) is a s e r i e s of positive t e r m s and is bounded, it converges. For an explicit calculation of V (x, w) in the case that w is an optimal strategy, we refer the reader to Section 2.3.' A strategy w is said to ~dominate strategy w if (a) v (x, w) .L v (x,w') (b) V (x, w) > V (x,w') for all x 1. I ; for some x 2 1 . zyxwvu zyx 113 GAMES O F ECONOMIC SURVIVAL That undominated strategies exist can be seen from the following argument. A withdrawal strategy w is completely characterized by the smallest integer N such that w(N) > 0, since the corporate account increases only by integral values and N acts as a "reflecting barrier." We shall call this N the N-value of w. NOW we have K (sn,w, t) = 0 for n = 1, 2, , N - x, so that, letting Vn (x, w) be the discounted amount in the withdrawal account out to time n, we have ... zyxwvu Now the t e r m in braces on the right is bounded by the same kind of argument that we used to show that V(x, w) was bounded. Since p 1, the factor pN-x becomes small if x is fixed and N becomes large. Hence all those strategies w whose N-values a r e sufficiently large a r e dominated by other strategies with smaller N-values. Since there a r e only a finite number of withdrawal strategies with bounded N-values, there a r e only a finite number of values V(x, w) to consider, and hence an undominated strategy w exists. Let @ be the set of undominated strategies. In Section 2.3. we shall characterize @ for a certain kind of one-person game of economic survival. zyxw zyxwvu c z 2.2. A Simple One-Person Survival Game Consider the game G for which al = 1, a2 = - 1, p1 > 0, p2 > 0, p1 + p a = 1, B = 0, and F = 0. We shall characterize the undominated strategies .for this game and thus obtain information about the result for the general one-person survival game. LEMMA 1: An undominated wo in G is a solution of the functional equation* (1) v(x,w 0 = max p p1 ~ ( - x1, wo) + p p 2 v ( x + I, w0j, max [a a21 + v(s - a, wo)] . *The optimal financial s t r a t e g i e s a r e not the only solutions of ( 1 ) . F o r instance, the function zyxwvutsrqpo s a t i s f i e s the equation V ( x ) = m a x [pp, V ( x a 20 - a - 1) t pp2 V ( x - a t 1 ) t a] , and we can r e g a r d this function a s the solution f o r a z e r o withdrawal strategy. But t h i s solution i s not economically interesting, s i n c e , although V (x)+cg, the expected discounted value of the withdrawal account f o r any finite length of t i m e i s z e r o . 506275 0 - 59 -4 114 zyxwvutsrqpo M. SHUBIK AND G. L. THOMPSON PROOF Equation (1) states merely that the player always has the choice of immediately playing the game o r first withdrawing an amount a and then playing. Moreover, he must choose the withdrawal payment a so that his value is maximized. It is obvious that any strategy, not a solution of ( l ) , is dominated by such a solution. zyxwvu zyxw zyxwvutsrq zyxwvu zyxwvuts COROLLARY: v [x, wo] is a monotone strictly increasing function of x. PROOF: From (1) we have V(x, wo) 1 1 + V(x - 1, wo), hence V(x, wo) > v ( x - 1, wo). Lemma 1 shows that the game is a dynamic programming problem in the sense of Bellman [l]. It also brings out a difficulty: namely, suppose that for one o r more values of a, both t e r m s of tine right-hand side of (1) a r e equal. An undominated strategy could then choose either term. We determine a unique choice by saying that he will not play the game when the t e r m s a r e equal. More precisely we give the following definition. DEFINITION By a special undominated strategy we shall mean an undominated strategy with the following property: wo (x) = 0 if, and only i f , ( 2) p p1 ~ ( - xI, wo) + p p2 V ( X+ 1, wo> > max [a + v (x - a, wo)l ail Remark: If p p1 < 1/2, the solution to the game is trivial, the player pays out all the resources to his withdrawal account immediately a s the game is "unfair" and not in his favor. We exclude this case and henceforth assume p p1 L 1/2. LEMMA 2: If wo is undominated, then there exists an x such that wo(x) > 0. PROOF: If wo(x) = 0 for all x, then, obviously, V(x,wo) = 0. But, since all a s s e t s can immediately be withdrawn, we see that V(x, wo) L x 2 0, which is a contradiction f o r positive x. Therefore for some x, wo(x) > 0. zy zyxwvu LEMNIA 3: Let wo be a special undominated strategy. If z is an x such that wo (x) = 0 and wo(x + 1) > 0, then wo(z + 1)= 1. PROOF: Suppose, on the contrary, that wo(z V (z + 1, wo) V(z, wo) + 12 V(z - + 1)= k + 1 > 1. 1, w0) +2 2.. By Lemma 1 we have . 2 V (z - k, wo) + k . Since the player always withdraws to produce the maximum amount possible, by Lemma 1 we have V(Z + I, wc') = V ( Z + 1 - n, wo) + n , hence all of the above inequalities a r e actually equalities. And, in particular, V(z, wo) = V(z - 1, wo) + 1. The definition of a special undominated strategy now shows that wo(z) > 0, contradicting the fact that wo(z) = 0. zyxwv zyxwvutsr zyxwvut 115 GAMES O F ECONOMIC SURVIVAL zyxwvuts zyxwvu LEMMA 4: Let wo be a special undominated strategy and let z be the smallest x such that wo (x + 1) > 0, then for all x 2 z we have wo (x) = x z. - PROOF No payment can be greater than x and V (x - a, w0) or - z, for if wo (x) = a > x - z then zzx -a + a L V (z,w0) + (x - z) , contradicting the fact that wo is a special undominated strategy and z is the smallest x such that wo(x + 1) > 0. (See (l).) If there is at least one x > z such that wo (x) c x - z, then there must be an x' > z such that wo(x') = 0, for otherwise (2) would be violated. (For instance, x' = x - wo(x) has t h i s property.) Let Z be the least x > z such that w(x) = 0. By an argument similar to that of Lemma 2, there is an x Z such that w(x) = 0 and w(x + 1) > 0. Call the smallest such zl. By an argument analogous to Lemma 3, w(zl + 1) = 1. Then we have the following information about wo: for x (z 0 w0 (x) = x-z forz<x<Z 0 f o r z z x 5 z1 1 f o r x = z1 + l . zyxw We shall show that wo is actualIy dominated by another strategy w' defined a s follows: :zyxwvuts ( 0 I x -z for x 5 z or if x = z1 for the first time, zyxwv for (i) z < x < Z -1 if x has ever been (ii) z 5 x ( Z w'(x) = 0 for (i)Z - 15x < (ii)Z 5 x 5 z1 wo(x) for x ? z1 + 1 i f x has always been < z1 , = z1 , z1 if x has always been < z1 , if x has ever been = z1 . . We shall show that strategy wo does not do as well as w' by comparing two games, one starting with C (0)= a and using strategy w', and the other starting with C ( 0 ) = a + 1 and using strategy wo, where F 5 a < a + 1 5 zl. Let C' (t) be the player's corporate account in the first game and Co(t) be his corporate account in the second game. We compare these two strategies for the same course of outcomes of individual plays of the game. 116 zyxwvu zyxwvu zyxwvu zyxwv M , SHUBIK AND G. L. THOMPSON Case 1: For some t 2 1, C' reaches z1 before Co reaches Z - 1. Then 1 is withdrawn from the Co account, but the C' remains unchanged so that at time t the two corporate accounts will be equal to zl. And since w' becomes the same a s wo after reaching zl, the two corporate accounts will be equal for the r e s t of the game. Hence we see that V ( a + 1, wo) - V(a, w') = pt < 1. zyxwvutsrq Case 2: For some t 2 1, Co reaches Z - 1 before C' reaches zl.. Then Z - 1 - z is withdrawn from Co and E - 2 - z is withdrawn from C' so that at time t the two corporate accounts will be equal (to z); and since w' is the same as wo when x 5 z, the two corporate accounts will remain equal for the r e s t of the game. Hence we s e e that V ( a + 1, wo) - V(a, w') = pt < 1. In both of the above cases it is false that V(a + 1, wo) 2 V(a, w') + 1, which must necessarily be the case for an undominated strategy. (One could do better than wo at a + 1 by first paying out 1 and then using w'.) From this contradiction we conclude that Z does not exist and wo(x) = x - z for x > z, concluding the proof. zyx zyxwvutsr THEOREM 1: There is a unique special undominated strategy wo defined as follows: t h e r e e x i s t s a z '0 suchthat w o ( x ) = O i f x < z and w o ( x ) = x - z if X L Z . PROOF: Observe from the definition of special undominated strategies that z (the largest x for which wo(x) = 0) is the same for all special undominated strategies. Lemmas 3 and 4 and mathematical induction now show that any two special undominated strategies a r e equal. Remarks: (A) If p = 1, then this one-person game becomes the classical gambler's ruin game. The unique special undominated strategy for this game would be to initially withdraw all funds and not play the game at all. (B) The calculation of the value of an example of a game of the type here discussed is carried out in Section 2.3. (C) If the changes of state a r e rational numbers al and a2, where a1 > 0 and a2 < 0, then the changes in the possible fortunes of the player will be multiples of d, where d is the greatest common divisor of al and a2. We regard the player a s ruined whenever his corporate account is less than / a 2 ( .The optimal strategy for the player is as follows: (a) initially pay out an amount Y = x - m 1 a2I where m is the largest integer such that Y is nonnegative; (b) there is a number z such that, if x ,< z, then wo(x) = 0, and if x 2 z, then wo (x)= x - z. (It can be shown in this case that x - z is a multiple of d.) 2.3 The Value of a Simple One-Person Game of Economic Survival In Section 2.2 we have established the form of the special optimal strategy w for a game with al = + 1 and a2 = - 1, and F = 0. The value of a game in which the special optimal strategy is used can be calculated directly as follows: Let z + 1 be the first value of the corporate account at which a payment is made to the withdrawal account under the optimal strategy wo. By setting PI = p and P2 = q = 1 - p, the value of the game is the solution to: GAMES O F ECONOMIC SURVIVAL zyxwv zyxw 117 v ( x , w O ) = p p v ( x + l , w O ) + p q v ( x1-, W O ) for x c z (3) subject to: ( 4) ( 5) V(0, wO) = 0 , zyxwvutsr zyxwvu zyxwvu v(z + 1, wO) = v(z, wO) + 1 . We set V(x, wo) = ax in ( l ) , which yields: Denoting the roots by a1 and cr2 , V(x, w0) = A1 CY? + A2 c ~ f. From (4) and (5) we obtain: and The solution may be expressed as: - V(x, wo) = A1 (2) zyxwv a!:} . The solutions for games with p p > 1/2 for different values of p and p have been calculated and examples a r e provided in the tables below. p = .995, p = .99 P = .7, p = .99 I 1 1 191.34 195.08 2 194.21 198.02 3 199.04 zyxw 195.08 198.02 p] 118 1 zyxwvuts 1.95 1.93 1 2.66 2.84 2.97 3.06 2 3.57 3.55 2 4.86 5.18 5.42 5.59 3 4.97 4.94 zyxwvuts zyxwvutsrq zyxwvutsrqpon M. SHUBIK AND G. L. THOMPSON P = .995,p = -55 P = .99,p = .55 I I Z zyxwv zyxwvu 6.71 7.15 7.48 7.71 4 8.28 8.83 9.24 9.52 5 9.65 10.28 10.75 11.09 6 10.85 11.57 12.10 12.47 7 11.94 12.73 13.32 13.72 13.80 14.43 14.87 15.48 15.95 3 I 4 6.20 6.16 5 7.33 7.27 6 8.38 8.32 7 19.391 9.32 8 9 1 10.31 8 9 10 10 The numbers in the boxes represent the values of the largest investment that will be made under an optimal policy. The underlined z ' s a r e the amounts associated with the optimal policies. 3.1. GENERAL DESCRIPTION OF TWO-PERSON GAMES OF ECONOMIC SURVIVAL A two-person game of economic survival is characterized by the following quantities: an m x n matrix A = Ila.. 11, a discount rate p (<l), ruin payments F1 and F2,ruin conditions 9 B1 and B2,corporate accounts C = (C1(t), C2(t)), and withdrawal accounts W = (W1(t),W2(t)). The role that each of these quantities plays is analogous to those played by the corresponding quantities in the one-person case. The subscript 1 r e f e r s to the first player and 2 r e f e r s to the second player. Let the initial amounts in the players' corporate accounts be C1(0) = x and C2(0) = y. We shall seek equilibrium point solutions for a simple class of these games for various values of x and y. 3.2. A Simple Two-Person Survival Game Here we consider the case in which B1= B2= 0, F1= F2 = F, and A is the matrix A = We shall characterize the equilibrium point solutions of this game for various initial fortunes x and y. In general it is false to assume that the players will always use optimal strategies for every matrix game A, as will be shown in Section 3.4. In the simple example selected here, however, the unique optimal strategies for each player are to choose each of his alternatives zyxwvu zyxwvu zyxwvu zyxw zyxwvutsrqp 119 GAMES O F ECONOMIC SURVIVAL with probability 1/2, and by known martingale theorems [2] there is no optimal way to deviate from these. Hence each player can expect +I and -1 with equal probability each time the game is played. Let us compute the values V (x, y) = (V1 (x, y), V2 (x, y)) of withdrawal accounts of each player, assuming that they begin with fortunes x and y and that they both always keep their fortunes in t h e i r corporate accounts. We must distinguish between two cases: the first, where at the termination of play the winner obtains only the "prize" F, and the second, where he obtains the prize F and adds the assets in h i s corporate account to h i s winnings to bring his total to x + y + F. Initially, we examine the first case. However, it is easier to give an economic interpretation to the second. This is done later in Section 3.4. For the first player the quantity V1 (x, y) satisfies (6) P v1 (x, y) =-v1 2 (x + 1, y - 1) +-P v1 (x - 1, y + 1) 2 4 By use of the trial solution V1 (x, y) = cXdYwhere c and d a r e constants, and the boundary conditions V1 (x + y, 0) = F and V2 (0, x + y) = 0, it can be shown that the solution to these difference equations is where If 0 < p < 1, then c1 > 1 and 0 < c2 < 1. (This is certainly the range of economic interest.) If x + y is relatively large, then the second factor in the denominator of (7) is negligible and we may write (9) F Vl(x, y) = -(1 c: - kX) where k = - c2 = c1 2 (k c2 < 1). zyxw zyxwvutsrq We use this approximate relation to investigate the solutions. Note that for fixed y the quantity V1 (x, y) is a monotone increasing, but bounded, function. Thus, given a fixed fortune y for player 2, player 1 will not want his capital account to be arbitrarily large, since it would be more profitable for him to withdraw some of this money and invest it elsewhere. Now suppose the players can withdraw any part of their corporate account, SO long as neither player is ruined. But if either player is ruined, the survivor gets F and forfeits the money in h i s corporate account. Player 1will withdraw a dollar whenever h i s expected value obtained by leaving it in is less than a dollar more than his expected value obtained by withdrawing it-in other words, when (for fixed y) x is such that 120 zyxwvutsr zyxwvuts zyxwvuts M. SHUBIK AND G. L. THOMPSON F -(1 - kX) + F 1 2-(1 CB - kX+l) . CB I 1 After some algebraic manipulations we take logarithms and obtain a relation of the form where b is a positive constant. Since we have made an approximation in the course of deriving the very simple formula (ll),we must expect our results from now on to be quantitatively inaccurate. However, for large discount rates (small discount factors) the e r r o r due to the approximations is small, so that our conclusions a r e qualitatively correct. Carrying out the analogous procedure for player 2 we find that, given a fortune x of player 1, the maximum fortune y that player 2 would keep in the game is given by the equation where the constant b is the s a m e as in equation (11). We a r e now in a position to analyze the equilibrium points in the game. 3.3. Equilibrium Points in the Two-Person Game I 1 ;\.-' \i zyxw zyxwvutsrqponmlk Figure 2 zyxwvutsrqponmlkj zyxwvutsrqpon ____----________ + j -_____ In Figure 1 we have sketched the graphs of equations (11) and (12) I 0 b y = b-ax 3-y and have numbered different areas of a the plane that lead to essentially difFigure 1 ferent behavior, described by a pair of strategies in equilibrium. In area 1 neither player has more money that he would like to have, based on the most pessimistic assumption that his opponent will not withdraw money from the game. It is impossible for them to move to a point at which either player would prefer to withdraw money rather than keep it in. Hence, i f the initial fortunes of the players specify a point A in the interior of 1, as the game progresses their fortunes move along a negative 45" line (see Figure 2) until \ 6' 121 GAMES O F ECONOMIC SURVIVAL zyxwvu zyxwvutsr zyxwvutsr zyxwvu the game terminates at one of the axes at which point one of the players is ruined. The equilibrium points also have maxmin properties in a r e a 1. If the initial fortunes specify a point B in the interior of area 2, then again neither player wishes to withdraw money. However, with probability 1, the play of the game will proceed until one of the players makes a withdrawal. This is shown in Figure 2 by the crossing of the negative 45" line with one of the lines of equation (11) o r (12). After one of the players withdraws part of his fortune, the point representing the joint fortunes in their corporate accounts moves back into area 2, and the game progresses until it eventually terminates at (0, b) or (b, 0). A possible path of play is illustrated in Figure 2. Let A be an initial fortune point in area 3 (see Figure 3). Here the second player has more money than he needs under a minmax assumption concerning his opponent's behavior. He withdraws down to the point A' which is in area 2. The game then proceeds as if it started at A' in area 2. In Figure 3, point B is in area 4 where both players a r e overcapitalized. Two possible equilibrium paths a r e as follows: first, player 2 may withdraw capital so that the fortune move to point B' which is in area 2 and the game proceeds from there; o r player 1 may withdraw to point B", after which player 2 withdraws to B"' in area 2 and the game proceeds from there. Other equilibrium paths between B' and B"' are also possible. Point C in Figure 3 represents a point in which player 2 has enough capital to make it rational for player 1 to withdraw from the game on a minmax assumption concerning player 2's behavior. This is shown at C". If, however, player 1 retained his capital, player 2 would withdraw down to C'. There is mixed strategy equilibrium involving the decisions of each player to keep in 211 h i s funds and the decision of player 1 to leave the game and player 2 to reduce investment. There a r e also many other equilibrium points. The equilibrium point solutions in areas 3', 4', and 6' a r e analogous to those in a r e a s 3, 4, and 6, but with the roles of the two players interchanged. Point A in area 5 of Figure 4 represents a point at which both players a r e overcapitalized. There a r e two obvious equilibrium points: in one player 2 withdraws to A' and then player 1 withdraws to A"'; in the other, player 1withdraws to A" and then player 2 withdraws to A"". There are many other equilibrium points. In particular, E is enforceable as follows: player 2 announces that he will withdraw to E' and, even though overcapitalized, will remain I 7 I zyxwvutsr zyxw - x Figure 3 50bZ75 0 - 59 - 5 Figure 4 x 122 zyxwvut zyxwvu zyxwvut M. SHUBIK AND G . L . THOMPSON zyxwvu there, regardless of what player 1 does; it is then rational for player 1 to withdraw to E, and the game proceeds from there. There is an analogous equilibrium point in which 1 withdraws to E" and then 2 withdraws to E. At point B in area 7 of Figure 4, player 2 is so overcapitalized that player 1 should quit entirely. If he does, the fortune point moves to B". On the other hand, if player 2 first withdraws his capital, the fortune point moves to B'; then player 1 should withdraw to B"' and the game proceeds in a r e a 2; there is a mixed strategy equilibrium involving these choices. The equilibrium point E is also enforceable in the same manner a s stated in the paragraph above. Many other equilibrium points a r e possible. The behavior in a r e a 7' is analogous to that in 7, with the roles of the players interchanged. In a r e a 8 both players a r e heavily overcapitalized and there are many equilibrium points, including E. The analysis is most satisfactory in a r e a s 1, 2, 3, and 3', since unique equilibrium points a r e obtained and these have maxmin properties. In other areas, the multiplicity of equilibrium points means that the outcome of the game will be settled by means of threats and counterthreats, in which case the equilibrium point analysis is not particularly conclusive. Experimental investigation of the course of play in areas 5 to 8 would be of interest. zyxwvutsrq 3.4. An Economic Interpretation of the One- and Two-Person Games The one-person game serves to demonstrate the "safety value" of liquid a s s e t s in a fluctuating market. It is analogous to an inventory problem. The penalty of being "out of stock" is the ruin of the firm. The next step in the investigation of models with greater realism involves having the rewards from the random walk depend upon the asset position of the firm. In the model in Section 2.2, the only use of the money in the corporate account was a s protection against ruin. It is possible to give a mathematical formulation for the policy of a firm which wishes to achieve and maintain a given level of safety prior to paying dividends. A discussion and examination of the effects of different policies is given elsewhere [4]. The two-person game has an economic analogue a market in which there is overcapacity with two firms present, but in which one firm could make a profit if it were the sole survivor. The zero-sum matrix indicates that if one firm makes a profit, the other must lose. For example, the only way in which business may be obtained is by obtaining the competitor's accounts. The prize, F, is the discounted value of the income obtainable by the surviving firm in the market. When one firm is ruined the value of the other is F plus its corporate a s s e t s at that time. In order to express this we would have had to express: zyxwvut V(X + y,O) = x + y + F. If x + y <<< F, then the same qualitative results a s above hold, with the linear equations (11) and (12) being replaced by nonlinear equations whose graphs are convex upward curves similar to the straight lines of Figure 1. If cooperation and side payments were permitted, the solution would be for the industry to "rationalize" immediately, with one f i r m exiting but being paid off by the other. An examination of the solution shows that the richer both firms are, the worse off they may be1 ! I zyxwvu zyxwvuts zyxwvu zyxwvuts GAMES O F ECONOMIC SURVIVAL V(x, x) L V(x + k, x + k) 123 . This phenomenon is caused by the prolongation of the length of the struggle for survival. If two financially weak firms compete, one will soon fail, leaving the other a lucrative market. If they a r e stronger, it will take a much longer time to reach a prosperous state, unless there is collusion and rationalization. 3.4. Comments and Further Problems It is evident that the investigation of the game with: zyxw by a simple change of units. If the matrix A is 2 x 2 and the one-period zero-sum game on A has a unique mixed strategy solution, then the players will always use optimal strategies in the matrix game each period. For A larger than 2 x 2, this is not necessarily so. A simple example demonstrates this: Let 1 1 1 1 The strategy (-, -, 0, 0) dominates (0, 0,- -) in the one period game but not neces2 2 2’ 2 sarily in the dynamic game in which the duration of play as well as the expected gain p e r period is of importance. A further discussion and economic interpretation of this type of game including cases where the one-period “subgame“ is nonzero sum is given elsewhere [4]. BIBLIOGRAPHY [l] Bellman, Richard, Dynamic Programming, Princeton University Press, 1957 [2] Doob, J. L., Stochastic Processes, Wiley, 1953 [3] Feller, W., The Theory of Probability, Wiley, 1950 [4] Shubik, Martin, Strategy and Markat Structure, Wiley, 1958 * * *