Monotone Comparative Statics for Games With Strategic Substitutes 1

Transcription

1 Monotone Comparative Statics for Games With Strategic Substitutes 1 By Sunanda Roy College of Business and Public Administration Drake University Aliber Hall Des Moines IA USA sunanda.roy@drake.edu And Tarun Sabarwal Department of Economics University of Kansas 1460 Jayhawk Blvd Lawrence KS USA sabarwal@ku.edu First Draft: November 2004 This Version: February 23, 2009 JEL Numbers: C60, C61, C62, C72 Keywords: Monotone comparative statics, Nonincreasing functions, Strategic substitutes, Parameterized games 1 We are grateful to Federico Echenique, Charles Moul, John Nachbar, Wilhelm Neuefeind, anonymous referees, and seminar audiences at Drake University, Iowa State University, Queen s University, University of Kansas, and Washington University in Saint Louis for helpful comments.

2 Monotone Comparative Statics for Games With Strategic Substitutes Abstract Under some conditions, parameterized games with strategic substitutes exhibit monotone comparative statics of equilibria. These conditions relate to a tradeoff between a direct parameter effect and an opposing, indirect strategic substitute effect. If the indirect effect does not dominate the direct effect, monotone comparative statics of equilibria are guaranteed. These conditions are available for best-response functions, differentiable payoff functions, and general payoff functions. Results are extended to payoff functions that may yield bestresponse correspondences. The analysis applies to asymmetric equilibria. Several examples are provided.

3 1 Introduction As is well-known, games with strategic complements and strategic substitutes are found in many areas of economics. 2 Although monotone comparative statics results for general games with strategic complements are well-developed, 3 results of similar generality are less commonly available for games with strategic substitutes. For example, we are not aware of a general result for such games that can be applied to show increasing equilibria in a simple, parameterized, asymmetric, Cournot duopoly with linear demand, constant marginal cost, and the standard product order on strategy spaces. Consider a linear inverse market demand curve given by p = a b(x 1 + x 2 ), where x 1 is the output of firm 1, and x 2 of firm 2. Suppose each firm has constant marginal cost c. Moreover, there is a subsidy of t c per unit, and this subsidy is split with an exogenously specified share 3 5 for firm 1, and share 2 5 for firm 2.4 Thus, firm 1 s marginal cost net of 2 Such games are defined in Bulow, Geanakoplos, and Klemperer (1985), and as they show, models of strategic investment, entry deterrence, technological innovation, dumping in international trade, natural resource extraction, business portfolio selection, and others can be viewed in a more unifying framework according as the variables under consideration are strategic complements or strategic substitutes. Additional classes of examples are provided by Cournot oligopolies, bargaining games (Nash demand game), and as described in Dubey, Haimanko, and Zapechelnyuk (2006), include games of team projects with complementary or substitutable tasks, and tournaments. 3 Some of this work can be seen in Topkis (1979), Lippman, Mamer, and McCardle (1987), Sobel (1988), Vives (1990), Milgrom and Roberts (1990), Zhou (1994), Milgrom and Shannon (1994), Milgrom and Roberts (1994), Shannon (1995), Villas-Boas (1997), Edlin and Shannon (1998), Echenique (2002), and Echenique and Sabarwal (2003), among others. Extensive bibliographies are available in Topkis (1998) and in Vives (1999). 4 Alternatively, the parameter t can be thought of as technological improvement, and ( 3 5, 2 5 ) can be thought 1

4 subsidy is c 3 t, and that of firm 2 is c 2 t. In this case, the unique equilibrium is given 5 5 by x (t) (x 1 (t), x 2 (t)) = (a c+(9 5 1)t 3b, a c+(2 9 5 )t 3b ). With the standard product order on strategy spaces, this example does not fit the framework of Milgrom and Shannon (1994), because the profit functions are not quasisupermodular. 5 Therefore, this game is not supermodular, and this example does not fit the framework of Topkis (1979), Sobel (1988), or Vives (1990). If the order on one of the strategy spaces is reversed, then it is known (see, for example, Milgrom and Shannon (1994), and Amir (1996)) that this example is a quasi-supermodular game. Of course, such a transformation can violate the single-crossing property in (player strategy; parameter), and standard results about monotone comparative statics might still not apply. 6 Moreover, asymmetric Cournot conditions rule out an application of Amir and Lambson (2000), and of the intersection point theorem of Tarski (1955). 7 of as differential adaptation of technological improvement. A slightly more general example is presented later. 5 Denote profit of firm 1 at (x 1, x 2, t) by f 1 (x 1, x 2, t), and consider the values a = 10, b = 1, c = 1, t = 0, and consider (x 1, x 2 ) = (3, 2), and (x 1, x 2 ) = (4, 3). Then, f1 (x 1, x 2, t) f 1 (x 1, x 2, t), but f 1 (x 1, x 2, t) < f 1 (x 1, x 2, t). 6 Indeed, as shown below, in slight variations of this game, it is easy to have the equilibrium strategy of either player increasing and that of the other player decreasing. 7 Tarski s intersection point theorem applies to linearly ordered spaces. It is noteworthy that one trick that can work for the special duopoly case is to compose the reaction functions of the two firms. This yields an increasing function. In this case, an equilibrium can be shown to exist, and at least for one of the players, equilibrium can be shown to be increasing, but (in asymmetric Cournot) not necessarily for the other player. Indeed, as shown below, it is easy to formulate examples of simple Cournot duopolies where the equilibrium is increasing for one player, and decreasing for the other. The same point applies to techniques that apply when the best-response of one player depends only on the aggregate best-response of other players. Of course, such techniques have been formulated primarily to prove existence theorems for Cournot oligopolies, 2

5 One general result is available for games with strategic substitutes. As shown by Villas- Boas (1997), in such games, equilibria do not decrease when a parameter increases. 8 As is well-known, and as highlighted again in simple examples here, this result cannot be strengthened to conclude existence of monotone comparative statics in general parameterized games with strategic substitutes. A more specialized result is available, too. When player strategy spaces are chains, symmetric equilibria are nondecreasing in the parameter. This can be inferred from results in, for example, Milgrom and Roberts (1994) and Villas-Boas (1997). Another derivation can be found in Roy and Sabarwal (2008). This paper presents conditions, which when satisfied, guarantee monotone comparative statics of equilibria in parameterized games with strategic substitutes. The results apply to general parameterized games with strategic substitutes and to asymmetric equilibria. The intuition behind these conditions is as follows. In either games of strategic complements, or games with strategic substitutes, there are two effects of an increase in the parameter. The direct effect arises because a parameter increase directly affects the payoff to each player. The indirect effect arises because a change in the parameter affects strategies of the other players, and strategic interaction among players indirectly affects a given player s strategy. In games with strategic complements, both effects work in the same direcand not necessarily to show increasing equilibria. See, for example, McManus (1964), Selten (1970), Roberts and Sonnenschien (1976), Bamon and Fraysee (1985), Novshek (1985), Kukushkin (1994), and Amir (1996), and additional discussion in Vives (1999). 8 Additionally, some aspects of non-monotone mappings that are increasing in some variables and decreasing in others are explored in Roy (2002). Also confer Roy and Sabarwal (2008) for another view of the result of Villas-Boas. 3

6 tion. An increase in the parameter directly increases marginal payoffs to each player, and serves to increase a player s strategy, and strategic complements imply that an increase in other player strategies serves to indirectly increase a given player s strategy. In games with strategic substitutes, these effects work in opposite directions. Similar to strategic complements, an increase in the parameter directly increases marginal payoffs to each player, and serves to increase a player s strategy, but in contrast to strategic complements, strategic substitutes implies that an increase in other player strategies serves to indirectly decrease a given player s strategy. We show that at a new parameter value, if this indirect effect does not dominate the direct effect, then a larger equilibrium exists. The conditions presented in this paper can be viewed as different ways to identify measures of this combined effect. Notice that as shown by Villas-Boas (1997), in the case of a Cournot oligopoly, if an analyst can choose a new partial order, then under certain conditions there exists a new partial order in which equilibria are increasing. The new partial order, however, might not necessarily be intuitive or relevant for natural parametric policy experiments. For example, for a Cournot oligopoly, the product order may be natural when considering the impact of taxes or subsidies on firm output, and the existence of some other partial order under which equilibria are increasing might not be relevant. In games of strategic complements, the product order is used commonly for the same reason; that is, to investigate the impact of different parameters on each agent s choice. The results here apply to cases where a partial order is considered as fixed. The paper proceeds as follows. Section 2 presents a simple example to provide some intuition regarding existence of monotone comparative statics, and to help interpret the more general results presented later. Section 3 presents general results that guarantee monotone 4

7 comparative statics of equilibria. These results are applicable to best-response functions that are nonincreasing in other player strategies, and therefore, apply to games with strategic substitutes. Section 4 characterizes a relationship between payoff functions and nonincreasing best responses. Section 5 defines a general parameterized game with strategic substitutes, and presents conditions on payoff functions that guarantee monotone comparative statics. Section 6 provides some examples. Finally, section 7 extends the results to the case where payoff functions may imply that best-responses are correspondences. 2 An Interpretive Example Example 1. Consider a standard Cournot duopoly with a linear inverse market demand curve given by p = a b(x 1 + x 2 ), where x 1 is output of firm 1, and x 2 of firm 2. Suppose each firm has constant marginal cost c. Moreover, there is a subsidy of t c per unit, and this subsidy is split with share ξ [0, 1] for firm 1, and share 1 ξ for firm 2. 9 Thus, firm 1 s marginal cost net of subsidy is c ξt, and that of firm 2 is c (1 ξ)t. In this case, the best-response function of firm 1 is g 1 (x 2, t) = a c+ξt bx 2 2b, and that of firm 2 is g 2 (x 1, t) = a c+(1 ξ)t bx 1. It is easy to check that g(x 2b 1, x 2, t) (g 1 (x 2, t), g 2 (x 1, t)) is a strictly decreasing function in (x 1, x 2 ), it is strictly increasing in t, and the unique equilibrium at t is x (t) (x 1 (t), x 2 (t)) = (a c+(3ξ 1)t, a c+(2 3ξ)t ). Consequently, 3b 3b ξ < 1 x 3 1 (t) is decreasing in t, and x 2 (t) is increasing in t, 1 ξ 2 x 3 3 1(t) is increasing in t, and x 2(t) is increasing in t, and 2 3 < ξ x 1(t) is increasing in t, and x 2(t) is decreasing in t. 9 The example in the introduction is the case where ξ =

8 This example shows the possibility of monotone comparative statics of asymmetric equilibria. For each ξ such that 1 ξ 2, and ξ 1, equilibrium is asymmetric and monotone nondecreasing in t. To gain some more intuition, consider first, the symmetric case; that is, ξ = 1. In this 2 case, as is obvious, an increase in t lowers marginal cost equally for each firm, and each firm s output goes up. Now consider the extreme case where firm 1 gets the entire benefit of an increase in t; that is, ξ = 1. In this case, when t increases, firm 1 s best response shifts out, but firm 2 s best response is unchanged. The implication is that firm 1 s output increases, while firm 2 s output goes down. Similarly, when ξ = 0, firm 2 gets the entire benefit of t, and its output increases with t, while firm 1 s output goes down. This example shows that although it may not be possible to guarantee monotone comparative statics in all games with strategic substitutes, nevertheless, some asymmetry may be consistent with monotone comparative statics in games with strategic substitutes. 3 Existence of Increasing Equilibria This section considers models in which the best-response functions are nonincreasing in endogenous variables, and nondecreasing in parameters. These properties hold in parameterized games with strategic substitutes. Consider the following sets of assumptions. Assumption I.A (X, ) is a nonempty, partially ordered set, T is a nonempty partially ordered set, When no confusion arises, the same symbol denotes the partial order on T. 6

9 and g : X T X is a function. For every x X, g(x, ) is nondecreasing in t. 11 For every t T, g(, t) is nonincreasing in x. 12 Assumpton I.B X is a nonempty, compact convex subset of a Banach space. For every t, g(, t) is continuous. Assumption I.C X is a nonempty, closed, bounded convex subset of a Banach space. For every t, g(, t) is a compact operator. A triple (X, T, g) is admissible if it satisfies either conditions I.A and I.B, or conditions I.A and I.C. 13 For each t, let FP(t) = {x X x = g(x, t)} be the fixed points of g at t. Schauder s theorem implies that for every t, FP(t) is non-empty. Notice that at this level, these assumptions are somewhat weaker than what are usually assumed in the monotone comparative statics literature. For example, there is no requirement that X be a lattice, that it be a product of individual strategy spaces, or it be endowed with a product order. 11 For every x, and for every t, ˆt T, t ˆt g(x, t) g(x, ˆt). 12 For every t, and for every x, y X, x y g(y, t) g(x, t). 13 Notice that assumption I.A is an integral component of a parameterized game with strategic substitutes, whereas assumption I.B or I.C is made to guarantee existence of an equilibrium via Brouwer-Schauder type theorems. 7

10 Notice further that in a parameterized game with strategic substitutes, each player s best response function is nonincreasing in other player strategies, and therefore, the product of the best-response functions of the players satisfies the nonincreasing-in-x property (in the product order). Similarly, when each player s best-response function is nondecreasing in the parameter t, then the product of the best-response functions of the players satisfies the nondecreasing-in-t property (in the product order). Theorem 1. Let (X, T, g) be an admissible triple. Fix t T, and let x FP(t ). Consider ˆt T such that t ˆt, and let ŷ = g(x, ˆt), and ˆx = g(ŷ, ˆt). If x ˆx, then there is ˆx FP(ˆt) such that x ˆx. Proof. Notice that x ŷ, because g is nondecreasing in t. Moreover, for every x in [x, ŷ], g(x, ˆt) [x, ŷ], and this can be seen as follows. Suppose x x ŷ. Then x ŷ implies that g(x, ˆt) g(ŷ, ˆt) x, where the first inequality follows from the fact that g(, ˆt) is nonincreasing, and the second follows from the condition in the theorem. Moreover, x x implies that g(x, ˆt) g(x, ˆt) = ŷ, where the inequality follows from nonincreasing g(, ˆt), and the equality follows from definition of ŷ. Therefore, the restriction of g(, ˆt) to [x, ŷ] is a map from [x, ŷ] to [x, ŷ]. By Schauder s theorem, there is ˆx [x, ŷ] such that g(ˆx, ˆt) = ˆx, and consequently, there is ˆx FP(ˆt) such that x ˆx. The intuition behind the condition in this theorem can be seen clearly in a two-player game, with players indexed i = 1, 2. Suppose player i s strategies lie in a non-empty, compact, convex interval X i of the real numbers, and there is a partially ordered parameter space T. Player i s best-response function is g i : X j T X i, with i j. For each i and t, suppose that g i (, t) is nonincreasing, and for each i, and for each x j X j, suppose that g i (x j, ) 8

11 is nondecreasing. Let X = X 1 X 2, and endow it with the product order (denoted ). Suppose g(x 1, x 2, t) (g 1 (x 2, t), g 2 (x 1, t)) is a continuous function in (x 1, x 2 ), and let FP(t) be the set of fixed points of g at t. 14 The application of the above theorem to this case is formalized in the following corollary. Corollary 1. Consider a game with two players, as above. Fix t T, let x = (x 1, x 2) FP(t ), and consider ˆt T with t ˆt. Let (ŷ 1, ŷ 2 ) = (g 1 (x 2, ˆt), g 2 (x 1, ˆt)), and let (ˆx 1, ˆx 2 ) = (g 1 (ŷ 2, ˆt), g 2 (ŷ 1, ˆt)). If x 1 ˆx 1 and x 2 ˆx 2, then there is ˆx = (ˆx 1, ˆx 2 ) FP(ˆt) such that x ˆx. The conditions in this corollary can be viewed as follows. Starting from an existing equilibrium, x = (x 1, x 2) at t = t, an increase in t has two effects on g 1 (, ). One effect is an increase in g 1, because best-response functions are nondecreasing in t. (This is a direct effect of an increase in t.) The other effect is a decrease in g 1, because an increase in t increases g 2 (x 2, t), and x 1 and x 2 are strategic substitutes. (This is an indirect effect arising from player 1 s response to player 2 s response to an increase in t.) Similar statements are valid for player 2 as well. Taken together, the conditions say that for each player, as long as the indirect strategic substitute effect does not dominate the direct parameter effect, there is a new equilibrium that is larger than x = (x 1, x 2 ). A simple graphical illustration of these conditions is presented in Figure 1. The intuition for the general case is similar. It is useful to note that if either condition in the corollary is not satisfied, this result may not necessarily apply. This can be seen graphically in figure 2, where the first condition is violated but the second condition is satisfied. An alternative figure can be constructed 14 Notice that this class of games allows for asymmetric and multiple equilibria. 9

12 x 2 g 1 (., ˆt) g 1 (., t ) ˆx 2 x 2 g 2 (., ˆt) g 2 (., t ) 0 x 1 ˆx 1 x 1 Figure 1: Existence of Increasing Equilibria similarly where the reverse is true. The idea of competing direct and indirect effects helps relate the conditions here to those that arise in models with strategic complements. In those models, the direct and indirect effects work in the same direction, and therefore, once a parameter increases, both effects serve to move the new equilibrium set higher. Moreover, in those models, once increasing equilibria have been demonstrated, additionally higher parameter values serve to increase equilibria further, and do not reverse any increases. When direct and indirect effects work in opposite directions, increasing equilibria are no longer guaranteed. Moreover, even when the tradeoff between indirect and direct effects implies a larger equilibrium at a higher parameter value, that tradeoff might not necessarily hold at additionally higher parameter values, and therefore, a demonstration of a favorable tradeoff at a parameter value does not necessarily 10

13 x 2 g 1 (., ˆt) g 1 (., t ) ˆx 2 g 2 (., ˆt) x 2 g 2 (., t ) 0 ˆx 1 x 1 x 1 Figure 2: Violation of first condition imply increasing equilibria at additionally higher parameter values. The following corollary to Theorem 1 is useful to exhibit increasing selections of equilibria. Corollary 2. Let (X, T, g) be an admissible triple. If for every x, g(g(x, t), t) is nondecreasing in t, then for every t ˆt, and for every x FP(t ), there is ˆx FP(ˆt) such that x ˆx. The following corollary presents a version of strong monotone comparative statics. It strengthens the result in Theorem 1 to show that all equilibria at ˆt are greater than x. Corollary 3. Let (X, T, g) be an admissible triple, with X a complete lattice. Fix t ˆt, { and let x FP(t ). Consider g(g(x, ˆt), ˆt), and let ˆx L = inf X x g(g(x, ˆt), ˆt) x }. If x ˆx L, then for every ˆx FP(ˆt), x ˆx, and If x ˆx L, then for every ˆx FP(ˆt), x ˆx. 11

14 Proof. Notice that g(g(x, ˆt), ˆt), is nondecreasing in x, and therefore, by Tarski s theorem, ˆx L exists, and is the smallest fixed point of g(g(x,ˆt), ˆt) at ˆt. Moreover, the set of fixed points of g(g(x, ˆt), ˆt) at ˆt is a complete lattice. The result now follows by noting that the set of fixed points of g at ˆt is a subset of the set of fixed points of g(g(x, ˆt), ˆt) at ˆt. Another condition that guarantees the conclusion of the second statement (strictly increasing equilibria) in this corollary is the following. If x ˆx L, and if FP(ˆt) is not a singleton, then for every ˆx FP(ˆt), x ˆx. To prove this statement, we use results in Dacic (1981), and Roy and Sabarwal (2008), which imply that in games with strategic substitutes, the equilibrium set is completely unordered; that is, no two elements are comparable. Therefore, if x ˆx L, and if FP(ˆt) is not a singleton, then x FP(ˆt), and the result follows. 4 A Monotonicity Theorem This section characterizes a relationship between objective functions and nonincreasing solutions in a general class of maximization problems. The result is of some interest in its own right. Its application here provides conditions on payoff functions that guarantee nonincreasing best responses. Recall that conditions on payoff functions that imply nondecreasing best responses are well documented in the literature, with one of the most general results given by the monotonicity theorem of Milgrom and Shannon (1994). Their result can be adapted to provide general conditions under which best responses are nonincreasing. 12

15 Recall from Milgrom and Shannon (1994) that when X is a lattice, 15 a function f : X R is quasi-supermodular if (1) f(x) f(x y) = f(x y) f(y), and (2) f(x) > f(x y) = f(x y) > f(y). Moreover, when X is a lattice and T is a partially ordered set, a function f : X T R satisfies single-crossing property in (x; t) if for every x > x and t > t, (1) f(x, t ) > f(x, t ) = f(x, t ) > f(x, t ), and (2) f(x, t ) f(x, t ) = f(x, t ) f(x, t ). Recall that the single-crossing property is an ordinal condition. Its cardinal version is increasing differences, with the well-known intuition that the function f(x, t) f(x, t) as a function of t crosses zero at most once, and only from below. We shall say that a function f : X T R satisfies decreasing single-crossing property in (x; t) if for every x > x and t > t, (1) f(x, t ) f(x, t ) = f(x, t ) f(x, t ), and (2) f(x, t ) < f(x, t ) = f(x, t ) < f(x, t ). Analogous to the singlecrossing property, the decreasing single-crossing property is an ordinal condition, and its cardinal version is decreasing differences, which implies that the function f(x, t) f(x, t) as a function of t crosses zero at most once, and only from above. This intuition is the motivation for our present terminology. 16 For an order on nonempty subsets of X, we use the standard (induced) set order used in the literature. That is, for non-empty subsets A, B of X, A B if for every a A, and for every b B, a b A, and a b B, where the operations, are with respect to. With these concepts, the proof of the monotonicity theorem in Milgrom and Shannon 15 This section uses standard lattice terminology. See, for example, Topkis (1998). 16 Amir (1996) proposes a similar dual single-crossing property in the context of one-dimensional strategies. 13

16 (1994) can be adapted to prove the following theorem. Theorem 2. Let X be a lattice, T be a partially ordered set, S be a nonempty subset of X, and f : X T R. Reverse the standard set order on X. 17 The following is true. arg max x S f(x, t) is monotone nonincreasing in (t, S), if, and only if, f is quasi-supermodular in x and satisfies decreasing single-crossing property in (x; t) Proof. For each (t, S), let M(t, S) = arg max x S f(x, t). ( =) Let t t, S S, x M(t, S), x M(t, S ). Consider x x. As x M(t, S), f(x, t) f(x x, t). Therefore, f(x x, t) f(x, t) = f(x, t) f(x x, t) f(x, t) f(x x, t) = f(x, t ) f(x x, t ), where the first implication follow from quasi-supermodularity, and the last implication follows from decreasing single-crossing property. Thus, x x M(t, S ). Similarly, as x M(t, S ), it follows that f(x, t ) f(x x, t ) = f(x x, t ) f(x, t ) f(x x, t ) f(x, t ) = f(x x, t) f(x, t), f(x x, t) f(x, t), where the first implication follow from quasi-supermodularity, and the third implication follows from decreasing single-crossing property. Thus, x x M(t, S). (= ) To show that f is quasi-supermodular in x, fix t T, and x, x X. Let S = 17 S is lower than S in the reverse order means that S S. 14

17 {x, x x }, and S = {x, x x }. Then S S. Suppose f(x x, t) f(x, t). Then x M(t, S). But then, as M(, ) is nonincreasing, it follows that x x M(t, S ), whence f(x, t) f(x x, t). The proof for strict inequality is similar. To show that f satisfies decreasing single-crossing property in (x; t), fix x x and t t. Let S = {x, x }. Suppose f(x, t) f(x, t). Then x M(t, S), and as M is nonincreasing, x = x x M(t, S). Consequently, f(x, t ) f(x, t ). The proof for strict inequality is similar. This theorem is interesting in its own right as characterizing nonincreasing solutions in a class of maximization problems. From the perspective of this paper, one main application of this theorem is the if direction, with a player s strategy space given by X, with S identically equal to X, and with strategy space of other players given by T. 5 Conditions on Payoff Functions Theorem 1 above can be viewed as a general result that can be applied to the product of best responses in a game with strategic substitutes. In this section, general parameterized games with strategic substitutes are defined, and conditions are presented on payoff functions that guarantee monotone comparative statics of equilibria. 5.1 Parameterized Games With Strategic Substitutes Consider a set of players I, with players indexed by i I. Each player i has a partially ordered strategy space (X i, i ). The overall strategy space is the product of X i, denoted X, 15

18 and endowed with the product order. 18 Consider a partially ordered set of parameters, T. Each player i has a payoff function, f i : X T R, denoted f i (x i, x i, t). The collection Γ = (I, T, (X i, i, f i ) i I ) is a parameterized game with strategic substitutes, if for every player i, (X i, i ) is a complete lattice, and f i is continuous, For every (x i, t), f i is quasi-supermodular in x i, For every x i, f i satisfies single-crossing property in (x i ; t), and For every t, f i satisfies decreasing single-crossing property in (x i ; x i ). Recall that the second and third items are standard ingredients of parameterized supermodular and quasi-supermodular games (as considered in Milgrom and Roberts (1990) and Milgrom and Shannon (1994)). Single-crossing property in (x i ; t) allows best responses to be nondecreasing in the parameter. In quasi-supermodular games, strategic complementarity is modeled by the assumption of single-crossing property in (x i ; x i ), and it implies that best responses are nondecreasing in other player strategies. We are interested in strategic substitutes, and the appropriate ordinal condition for that is the decreasing single-crossing property in (x i ; x i ), as defined above. As shown by Theorem 2 above, with this definition, each player s best response is nonincreasing in other player strategies. The next subsection presents conditions on differentiable payoff functions that guarantee monotone comparative statics, and the subsection after that presents conditions on general 18 The topology on X i is the standard order interval topology, and the topology on X is the product topology. 16

19 payoff functions. 5.2 Differentiable Payoff Functions Let Γ = (I, T, (X i, i, f i ) i I ) be a parameterized game with strategic substitutes, and suppose the number of players is finite, indexed i = 1,...,I. Suppose further that for each (x i, t), arg max xi X i fi (x i, x i, t) is singleton-valued, 19 and for each i, f i is C 2. Case I. One-dimensional strategies and parameters Suppose for each i = 1,..., N, X i R, and T R. In this case, the condition in Theorem 1 translates to a condition on payoff functions, as follows. For player i = 1, the condition in the Theorem 1 can be written as ( g 1 (g 2 (x 2, t), g 3 (x 3, t),...,g N (x N, t), t) ) t (x,t ) 0, and for each player i = 2,..., N, a similar condition with an appropriate change of index. 20 Using the implicit function theorem, it is easy to calculate that ( g 1 (g 2 (x 2, t),...,g N (x N, t), t) ) t (x,t ) 0 f 1 1,t + N n=2 f 1 1,n ( ) fn n,t fn,n n (x,t ) 0. Here, a superscript on a payoff function f indexes a player, and subscripts denote partial derivatives. Thus, for example, f i i,n = 2 f i x i x n, and f i i,t = 2 f i x i t This is true when f i is strictly quasi-concave in x i. Alternatively, we may assume that f i is strictly concave in x i. 20 As usual, to apply this version, we suppose that the derivative is well-defined; in particular, (x, t ) is in the interior. 21 Notice that in a parameterized game of strategic substitutes, for n i, f i i,n < 0 formalizes strategic substitutes, and for n = i, f i i,n < 0 formalizes strict concavity in own argument. Moreover, fi i,t > 0 formalizes increasing differences in t. 17

20 With an appropriate change of index, it follows that if for every i = 1,..., N, f i i,t + n i f i i,n then the condition in Theorem 1 is satisfied. ( ) fn n,t 0, fn,n n (x,t ) These conditions are a natural analogue to those in Theorem 1. For a given player i, the term fi,t i is positive, and captures the direct effect of an increase in t on i s marginal profits, fi. i The term ( ) fi,n i fn n,t is negative, and captures the indirect effect of an fn,n n n i increase in t. This indirect effect can be viewed as the sum of indirect effects, one for each ( ) competitor player n i, where each competitor player s effect is given by fi,n i fn n,t. In fn,n n ( ) this last expression, fn n,t measures the (positive) change in player n s best-response from fn,n n an increase in t, and f i i,n measures the (negative) change in i s marginal profits from an increase in n s best-response. As earlier, if the indirect effect does not dominate the direct effect, monotone comparative statics are guaranteed. The transparency and ease of use of this condition can be seen in several examples provided later. Case II. Multi-dimensional strategies and parameters Suppose for each i = 1,...,N, X i R N i, and T R N 0. Again, using the implicit function theorem, for player i = 1, the relevant condition is ( D t g 1 (g 2 (x 2, t),..., g N (x N, t), t) ) (x,t ) [ = [D 1 f1 1] 1 D t f1 1 + )] n=2d N n f1 1 ( [D nfn n] 1 D t fn (x n.,t ) For reference, notice that the dimension of the matrix on either side of the inequality is 18

21 N 1 N 0. When N 0 = 1, [ N [D 1 f1 1 ] 1 D t f1 1 + ( ) ] D n f1 1 [Dn fn n ] 1 D t fn n 0 n=2 (x,t ) is a vector inequality (each component is greater than or equal to 0). Similarly, when N 0 > 1, we want N ( ) [D 1 f1] [D ] 1 1 t f1 1 + D n f1 1 [Dn fn] n 1 D t fn n 0, n=2 (x,t ) that is, the linear operator on the left hand side is a positive operator. If a similar condition holds for each i, then the condition in Theorem 1 is satisfied. 5.3 General Payoff Functions Let Γ = (I, T, (X i, i, f i ) i I ) be a parameterized game with strategic substitutes. Suppose further that for each i, for each sub-complete, sub-lattice S X i, and for each (x i, t), arg max xi S f i (x i, x i, t) is singleton-valued. 22 Let g i (x i, t, S) = arg max xi S f i (x i, x i, t), and g = (g i ) i I. (When S = X i, we shall find it convenient to suppress the notation for S.) Consider the following definitions. A nondecreasing selection ˆx is a nondecreasing function ˆx : T X. 23 Let ˆx be a nondecreasing selection. A payoff function f i satisfies monotone comparative statics relative to ˆx, if the function (x i, t) f i (x i, ˆx i (t), t) is quasi-supermodular in x i and satisfies the single-crossing property in (x i ; t). A payoff function f i satisfies monotone comparative statics, if for every nondecreasing selection ˆx, f i satisfies monotone comparative statics relative to ˆx. The following results are immediate consequences of the monotonicity theorem of Milgrom and Shannon (1994). 22 This is true when f i is strictly quasi-concave in x i. 23 t ˆt = ˆx(t) ˆx(ˆt). 19

22 Lemma 1. Let ˆx be a nondecreasing selection. g i (ˆx i (t), t, S) is nondecreasing in (t, S), if, and only if, f i satisfies monotone comparative statics relative to ˆx. Lemma 2. For every nondecreasing selection ˆx, g i (ˆx i (t), t, S) is nondecreasing in (t, S), if, and only if, f i satisfies monotone comparative statics. Notice that in the above definitions, quasi-supermodularity in x i is not an additional restriction; it is already present in the definition of a parameterized game with strategic substitutes. The main condition is single-crossing property in (x i ; t), where f i is evaluated at (x i, ˆx i (t), t). This takes care of the joint (direct and indirect) effect of a change in t, relative to ˆx. The direct effect of the parameter change is captured directly by a change f i due to a change in t, whereas the indirect effect of the parameter is captured by the change in f i due to a change in ˆx i (t). Say that a parameterized game with strategic substitutes satisfies monotone comparative statics property, if for every i, f i satisfies monotone comparative statics property. Say that a parameterized game with strategic substitutes has monotone comparative statics of equilibria, if for every t ˆt in T, there exists x FP(t ), and there exists ˆx FP(ˆt), such that x ˆx. The lemmas above allow us to prove the following theorem. Theorem 3. Let Γ be a parameterized game with strategic substitutes, and let each f i be strictly quasi-concave in x i. If Γ satisfies monotone comparative statics property, then it has monotone comparative statics of equilibria. Proof. Fix t ˆt. By Schauder s theorem, there is x FP(t ). Consider the nondecreasing selection, ˆx(t) = g(x, t). Notice that x = ˆx(t ). Moreover, by the lemma above, for every 20

23 i, g i (ˆx i (t), t) is nondecreasing in t. Thus, for t ˆt, x = g(ˆx(t ), t ) g(ˆx(ˆt), ˆt) = g(g(x, ˆt), ˆt), as required by Theorem 1. Thus, there is ˆx FP(ˆt) such that x ˆx. Notice that the assumption of each f i satisfying monotone comparative statics for every nondecreasing selection ˆx(t) is not necessary. For the theorem to hold, each f i is required to satisfy monotone comparative statics relative to a particular nondecreasing selection only; the one given by ˆx(t) = g(x, t). Therefore, in applications, knowledge or computability of best responses can increase potential applications of this theorem. 6 Examples Example 1 (continued). Consider the Cournot duopoly example presented earlier. The profit of each firm is given by f 1 (x 1, x 2, t) = (a b(x 1 + x 2 ))x 1 (c ξt)x 1, and f 2 (x 1, x 2, t) = (a b(x 1 + x 2 ))x 2 (c (1 ξ)t)x 2. In order to apply the conditions presented for differentiable payoff functions, it is easy to calculate that f1,2 1 = b, f1 1,t = ξ, f2 2,2 = 2b, f2 2,t = 1 ξ. Consequently, f 1 1,t + f 1 1,2 f 2 2,t + f 2 1,2 ( 2,t (x > 0 ξ > f2,2) 1, and similarly, 2,t 3 ) ( 1,t (x > 0 ξ < f1,1) 2. 1,t 3 ) It follows that equilibria increase in the range ξ ( 1, 2 ), as earlier

24 Example 2. Consider a common pool resource game. 24 Consider two players, indexed i = 1, 2, each with an endowment e > There are two investment options a common resource (such as a fishery) that exhibits diminishing marginal return, and a (potentially asymmetric) outside option with constant marginal return. If player i invests an amount x i e of his endowment into the common resource, he receives a proportional share of the total output. Payoff to player i is given by f i (x 1, x 2 ) = r i (e x i ) + x i x 1 + x 2 ( a(x1 + x 2 ) b(x 1 + x 2 ) 2). Here, r i > 0 is player i s potentially asymmetric marginal return from the outside option, x i x 1 +x 2 is player i s proportional share in common pool resource output, and total output of the common pool resource is (a(x 1 + x 2 ) b(x 1 + x 2 ) 2 ). It is well-known that in a Nash equilibrium, the common pool resource is over-appropriated, that is, total individual investment is greater than socially optimal investment. This has a long run implication for conservation of the resource. A question of interest is whether a regulator can provide decentralized tax incentives to induce both players to reduce their individual investment in the common pool resource, and thereby reduce over-appropriation. Equivalently, under what conditions will a subsidy increase individual investment in the common pool resource? Consider the following subsidy-parameterized payoff for player i. f i (x 1, x 2, t) = (1 t)r i (e x i ) + 24 See, for example, Ostrom, Gardner, and Walker (1994). x i x 1 + x 2 ( a(x1 + x 2 ) b(x 1 + x 2 ) 2). 25 It can be shown that the outcome does not depend on e, and therefore, e is taken to be uniform across players. 22

25 It is easy to calculate that f 1 1,t = r 1, f 1 1,2 = b, f2 2,t = r 2, f 2 2,2 = 2b. Consequently, f 1 1,t + f 1 1,2 ( f2 2,t f 2 2,2) (x,t ) 0 2r 1 r 2. Similarly, it can be shown that f 2 2,t + f 2 1,2 ( f1 1,t f 1 1,1) (x,t ) > 0 r 2 r 1 2. It follows that equilibria increase in the cone determined by r 2 = 2r 1 and r 2 = 1 2 r 1 in the (r 1, r 2 ) space. Example 3. Consider a game of team projects with substitutable tasks. 26 Suppose a project is to be accomplished by a team of N 2 players, each choosing task (or effort) x i [0, 1], with probability of success x i. The quadratic cost of effort x i is c i 2 x 2 i, and is allowed to be asymmetric across players. Tasks are substitutable in the sense that each player by herself can make the project successful. The probability of success is 1 n j=1 (1 x j). If the project is successful, player i receives a parameterized reward r(t) > 0 (with 0 t T, and r (t) > 0.) 27 Otherwise, the player receives zero. Therefore, the payoff to player i is f i (x 1,...,x N, t) = r(t)(1 n (1 x j )) c i 2 x2 i. 26 This version is based on Dubey, Haimanko, and Zapechelnyuk (2006). 27 The parameter t can be viewed as technological improvement, or subsidy provided, or reward provided to j=1 induce an increase in effort (or probability) of task completion. As shown in the example, the best response function depends on r(t) c i, where c i measures player i s costs, and therefore, r(t) can be viewed as a reward enhancement parameter relative to a player s costs. 23

26 A question of interest is when does an increase in reward increase each team member s effort to complete the task? It is easy to calculate that the best response of player i is x i = r(t) c i j=1,...,n;j i (1 x j ), and this best response is nonincreasing in other player actions, and nondecreasing in t. Notice that for player i = 1, f 1 1,t = r (t) n j=2 (1 x j) = r (t) r(t) c 1x 1, f 1 1,1 = c 1, and for n 1, f 1 1,n = r(t) j=2,...,n;j n (1 x j). A similar calculation can be made for the other players. Therefore, f 1 1,t + N f1,n 1 n=2 ( fn n,t f n n,n) (x,t ) = r (t) r(t) c 1x 1 + N n=2 r(t) 1 x n = r (t) r(t) c 1x 1 [ 1 N n=2 x n 1 x n N j=2 (1 x j )r (t) r(t) x n A similar calculation can be made for the other players. Notice that if for every n, x n 1 N, ]. then N n=2 x n 1 x n N n=2 1 N N = 1. Therefore, if for every player n, N 1 x n 1, then the N equilibrium increases in the parameter. Notice this bound is independent of a particular profile of strategies, and therefore, it applies regardless of the particular equilibrium under consideration. Example 4. Consider a game of tournaments. 28 Suppose a tournament has 3 players, where a parameterized reward r(t) (with 0 t T, and r (t) > 0) 29 is shared by the players 28 This version is based on Dubey, Haimanko, and Zapechelnyuk (2006). 29 It can be shown that the best-response function depends on r(t) c i, where c i measures player i s costs, and therefore, r(t) can be viewed as a relative reward enhancement parameter, relative to a player s costs. 24

27 who succeed in the tournament. If one player succeeds, he gets r(t) for sure, if two players succeed, each gets r(t) with probability one-half, and if all players succeed, each gets r(t) with probability one-third. Each player chooses effort x i [0, 1] with probability of success x i. Expected reward per unit for player i is π i (x i, x j, x k ) = x i (1 x j )(1 x k ) x ix j (1 x k ) x ix k (1 x j ) x ix j x k. The quadratic cost of effort x i is c i 2 x 2 i, and is allowed to be asymmetric across players. The payoff to player i is expected reward minus cost of effort. That is, f i (x i, x j, x k, t) = r(t)π i (x i, x j, x k ) c i 2 x2 i, In this case a question of interest is the following: if tournament organizers increase reward, when will all players compete more strongly? 30 Notice that f i i,j = r(t)π i i,j(x k ), f i i,k = r(t)π i i,k (x j), f i i,t = r (t)π i i(x j, x k ), fj j,t f j j,j = r (t)π j j (x i,x k ) c j, fk k,t f k k,k = r (t)π k k (x i,x j ) c k, x i = r(t )π i i (x j,x k ) c i, x j = r(t )π j j (x i,x k ) c j, x k = r(t )π k k (x i,x j ) c k. Therefore, fi,t i + fi,j( i fj j,t ) + f f j i,k i ( fk k,t j,j fk,k) k (x,t ) > 0 r (t)π i i(x j, x k ) + r (t)π i i,j(x k )x j + r (t)π i i,k (x j)x k (x,t ) > 0 π i i (x j, x k ) + πi i,j (x k )x j + πi i,k (x j )x k > Tournament organizers might have an incentive to have players compete more strongly, perhaps because it increases audience size, and therefore, ticket sales. 25

28 Moreover, π i i (x j, x k ) + πi i,j (x k )x j + πi i,k (x j )x k = (1 x j )(1 x k ) ( x j (1 x k ) + x k (1 x j )) x j x k + [ 1 3 x k 1 ((1 2 x k ) + x k )] x j + [ 1 3 x j ( )] 1 2 (1 x j ) + x j x k = (1 x j)(1 x k ), implies that the condition on payoff functions is satisfied, if each of x i, x j, x k is less than 1. In other words, if the equilibrium is not degenerate, that is, no player wins the tournament for sure, then the equilibrium increases with the parameter. The result extends to N player tournaments. The notationally intensive details for that case are provided in the appendix. Example 5. Consider a modified version of a generalized Nash demand game. 31 There are two players, indexed i = 1, 2. Player i bids for a share x i of a total pie, the size of which is given by x 1 + x 2. The probability the pie is split as bid is given by p(s), where s = x 1 + x 2, with the probability going down with the sum of the bids, that is, p (s) < 0. To encourage bidding for higher rewards, consider a potentially asymmetric parameterized incentive to bid high, given by t i x i. In other words, payoffs 32 are given by f 1 (x 1, x 2, t 1 ) = p(s)x 1 + t 1 x 1, and f 2 (x 1, x 2, t 2 ) = p(s)x 2 + t 2 x See, for example, Schipper (2003). 32 Another interpretation of the example is that academic researchers are bidding for outside grants, and a university provides an incentive by matching the grant amount. The matching proportion (perhaps 0 t i 1) can be potentially asymmetric across universities, or across researchers. A question of interest is, if matching proportions are increased, when do all researchers increase their bids, even though the probability of success goes down. 26

29 It can be calculated that f1,t 1 1 = 1 = f2,t 2 2 f1,2 1 = p (s)x 1 + p (s) f2,2 2 = p (s)x 2 + 2p (s) It is easy to see that the appropriate condition for monotone comparative statics for player 1 is f 1 1,t 1 + f 1 1,2 ( f2 2,t 2 f 2 2,2 ) (x,t ) 0 p (s )x 1 +p (s ) p (s )x 2 +2p (s ) 1 x 2 x 1 p (s ) p (s ) (assuming f 2 1,2 0 and p (s ) > 0). Consider the exponential family of densities; p(s) = αe sα, with α, s > 0. It can be calculated that p (s) < 0, p (s) > 0, p (s) = 1, and p (s) α f2 1,2 0 x 2 1. Similar calculations can α be made for player 2. Consequently, monotone comparative statics exist if at the original equilibrium, x 1, x 2 1, and α x 1 x 2 1. As the median of an exponential is smaller than α the mean, this allows considerable asymmetry in outcomes to be consistent with monotone comparative statics. No doubt, similar applications of these results can be made to other games as well. In particular, an application of these results does not require knowledge of best-response functions, or closed form solutions for an equilibrium. Therefore, from a practical point of view, the result here can have broad applications. 7 Some Extensions This section extends Theorem 1 and Theorem 3 above to the case where best-responses may be multi-valued. 27

30 7.1 Condition on Correspondences This subsection extends Theorem 1 to the case where best responses may be correspondences. 33 Consider the following definitions. Assumption II.A (X, ) is a nonempty, complete lattice, T is a nonempty partially ordered set, 34 and g : X T X is a correspondence. For every x X, g(x, ) is nondecreasing in t. 35 For every t T, g(, t) is nonincreasing in x, and nonempty-sublattice-valued. 36 Assumpton II.B X is a nonempty, compact, convex subset of a Banach space. For every t, g(, t) is upper hemi-continuous, and nonempty-compact-convex valued. Assumption II.C X is a nonempty, closed, convex subset of a Banach space. For every t, g(x, t) is relatively compact. 33 This subsection uses standard lattice terminology. See, for example, Topkis (1998). Moreover, as earlier, for an order on nonempty subsets of X, we use the standard (induced) set order used in the literature. For non-empty subsets A, B of X, A B if for every a A, and for every b B, a b A, and a b B, where the operations, are with respect to. 34 When no confusion arises, the same symbol denotes the partial order on T. 35 For every t, ˆt T, t ˆt for every x, g(x, t) g(x, ˆt). 36 For every x, y X, x y for every t, g(y, t) g(x, t). 28

31 For every t, g(, t) is upper hemi-continuous, and nonempty-closed-convex-valued. A triple (X, T, g) is admissible-ii if it satisfies either conditions II.A and II.B, or conditions II.A and II.C. 37 For each t, let FP(t) = {x X x g(x, t)} be the fixed points of g at t. Theorems of Kakutani-Glicksberg-Ky Fan, or Bohnenlust-Karlin imply that for every t, FP(t) is non-empty. As earlier, although for assumptions II.A II.C, there is no assumption of a product order on X, the main application we have in mind is for a game with strategic substitutes, where each player s best-response correspondence is nonincreasing in other player strategies, and therefore, the product of the best-response correspondences of the players satisfies the nonincreasing-in-x property (in the product order). Similarly, when each player s bestresponse correspondence is nondecreasing in t, the product of the best-response correspondences of the players satisfies the nondecreasing-in-t property (in the product order). The following extends Theorem 1 to correspondences. 38 Theorem 4. Let (X, T, g) be an admissible-ii triple. Fix t T, and let x FP(t ). Consider ˆt T such that t ˆt, and let ŷ = sup X g(x, ˆt). If x inf X g(ŷ, ˆt), then there is ˆx FP(ˆt) such that x ˆx. Proof. Notice that x ŷ, because g is nondecreasing in t, (hence g(x, t ) g(x, ˆt),) x g(x, t ), and sup g(x, t ) ŷ. Moreover, for every x in [x, ŷ], g(x, ˆt) [x, ŷ], and 37 As earlier, notice that assumption II.A is an integral component of a parameterized game with strategic substitutes, whereas assumption II.B or II.C is made to guarantee existence of an equilibrium via Kakutani- Glicksberg-Ky Fan, or Bohnenlust-Karlin, respectively. 38 We are grateful to an anonymous referee for pointing out an earlier version of this extension of Theorem 1. 29

32 this can be seen as follows. Suppose x x ŷ. Then x ŷ implies that inf g(x, ˆt) inf g(ŷ, ˆt) x, where the first inequality follows from the fact that g(, ˆt) is weakly decreasing with respect to, and the second follows from the condition in the proposition. Moreover, x x implies that sup g(x, ˆt) sup g(x, ˆt) = ŷ, where the inequality follows from weakly decreasing g(,ˆt), and the equality follows from definition of ŷ. Therefore, the restriction of g(, ˆt) to [x, ŷ] is a correspondence from [x, ŷ] to [x, ŷ]. By Kakutani-Glicksberg-Ky Fan or by Bohnenlust-Karlin, there is ˆx [x, ŷ] such that ˆx g(ˆx, ˆt). Consequently, there is ˆx FP(ˆt) such that x ˆx. 7.2 Condition on General Payoff Functions This subsection extends Theorem 3 to include payoff functions that may yield best responses that are correspondences. Consider the following definitions. The definitions in this paragraph are from Shannon (1995). For non-empty subsets A, B of X, A is strongly lower than B, denoted A s B if for every a A, and for every b B, a and b are ordered, and a b A, and a b B. For non-empty subsets A, B of X, A is completely lower than B, denoted A c B if for every a A, and for every b B, a b. A correspondence φ : T X is strongly nondecreasing in t, if for every t ˆt, φ(t) s φ(ˆt), and for every t ˆt, φ(t) c φ(ˆt). For t T, and nonempty S X, a correspondence φ, mapping (t, S) φ(t, S) is strongly nondecreasing in (t, S), if for every t ˆt and S Ŝ, φ(t, S) s φ(ˆt, Ŝ), and for every t ˆt and S Ŝ, φ(t, S) c φ(ˆt, Ŝ). Let Γ = (I, T, (X i, i, f i ) i I ) be a parameterized game with strategic substitutes. Suppose further that for each i, for each sub-complete, sub-lattice S X i, and for each (x i, t), 30

33 arg max xi S f i (x i, x i, t) is convex-valued. 39 Let g i (x i, t, S) = arg max xi S f i (x i, x i, t), and g = (g i ) i I. 40 Let φ : T X be a strongly nondecreasing correspondence. A payoff function f i satisfies monotone comparative statics-ii relative to φ, if for every selection ˆx : T X from φ, the function (x i, t) f i (x i, ˆx i (t), t) is strictly quasi-supermodular in x i, 41 and satisfies strict single-crossing property in (x i ; t). 42 A payoff function f i satisfies monotone comparative statics-ii, if for every strongly nondecreasing correspondence φ, f i satisfies monotone comparative statics-ii relative to φ. The following result is an immediate consequence of theorem 4 in Shannon (1995). Lemma 3. Let φ : T X be a strongly nondecreasing correspondence. For every selection ˆx : T X from φ, g i (ˆx i (t), t, S) is strongly nondecreasing in (t, S), if, and only if, f i satisfies monotone comparative statics-ii relative to φ. Say that a parameterized game with strategic substitutes satisfies monotone comparative statics-ii property, if for every i, f i satisfies monotone comparative statics-ii. Say that a parameterized game with strategic substitutes has monotone comparative statics of equilibria-ii, if for every t ˆt in T, there exists x FP(t ), and there exists ˆx FP(ˆt), such that x ˆx. The lemma above allows us to prove the following theorem. Theorem 5. Let Γ be a parameterized game with strategic substitutes, and let each f i be 39 This is true when f i is quasi-concave in x i. 40 When S = X i, we shall find it convenient to suppress the notation for S. 41 As in Shannon (1995), a function f : X R is strictly quasisupermodular, if for all unordered x, y X, f(x) f(x y) = f(x y) > f(y). 42 As in Shannon (1995), a function f : X T R satisfies strict single-crossing property in (x; t), if for every x x and t > t, f(x, t ) f(x, t ) = f(x, t ) > f(x, t ). 31