Ordinal potentials in smooth games

In the class of smooth non-cooperative games, exact potential games and weighted potential games are known to admit a convenient characterization in terms of cross-derivatives (Monderer and Shapley in Games Econ Behav 14:124–143, 1996a). However, no analogous characterization is known for ordinal potential games. The present paper derives necessary conditions for a smooth game to admit an ordinal potential. First, any ordinal potential game must exhibit pairwise strategic complements or substitutes at any interior equilibrium. Second, in games with more than two players, a condition is obtained on the (modified) Jacobian at any interior equilibrium. Taken together, these conditions are shown to correspond to a local analogue of the Monderer–Shapley condition for weighted potential games. We identify two classes of economic games for which our necessary conditions are also sufficient.


Introduction
In a potential game (Rosenthal 1973;Monderer and Shapley 1996a), players' preferences may be summarized in a single common objective function. 1 Knowing whether a specific game admits a potential can be quite valuable. For example, the existence of a potential reduces the problem of finding a Nash equilibrium to a straightforward optimization problem. Therefore, it is of some interest to know the conditions under which a potential exists. Clearly, sufficient conditions are most desirable. However, necessary conditions are also important. After all, such conditions may help avoiding a futile search for a potential. Moreover, as will be shown, necessary conditions may be indicative of sufficient conditions as well. 2 This paper considers smooth games, i.e., non-cooperative n-player games with the property that strategy spaces are non-degenerate compact intervals and payoff functions are twice continuously differentiable (Vives 1999). In the class of smooth games, we derive simple necessary conditions for the existence of a generalized ordinal potential. 3 Certainly, a generalized ordinal potential cannot exist if it is possible to construct a strict improvement cycle (Voorneveld 1997), i.e., a finite circular sequence of strategy profiles with the property that moving to the next profile in the sequence amounts to one player strictly raising her payoff by a unilateral change in strategy. However, in general, identifying strict improvement cycles in smooth games is not straightforward. We therefore consider a specific path in the neighborhood of a fixed strategy profile x * N , and shrink it to virtually infinitesimal size, as if using a pantograph. 4 This differential approach leads to simple necessary conditions for the local feasibility of a generalized ordinal potential. In particular, it constitutes the basis of our main result, a local analogue of the Monderer-Shapley condition for the existence of an exact (or weighted) potential in smooth games. 5 1 3

Ordinal potentials in smooth games
The analysis starts by considering strict improvement cycles that involve two players only. In this case, the existence of a generalized ordinal potential is shown to imply that the product of the slopes of any two players' mutual local best-response functions (or, more generally, the product of the corresponding cross-derivatives) must be nonnegative at any strategy profile at which at least two first-order conditions hold in the interior. Thus, borrowing the terminology familiar from contributions such as Bulow et al. (1985), Amir (1996), Dubey et al. (2006), and Monaco and Sabarwal (2016), we obtain as our first main necessary condition that the game must exhibit pairwise strategic substitutes or complements at any such point.
Next, we consider strict improvement cycles that involve more than two players. In the simplest case, the path runs along the edges of a small rectangular box that contains a fixed strategy profile x * N at its center. To formalize the resulting condition on crossderivatives, we introduce a modified Jacobian at x * N by replacing all diagonal entries of the Jacobian of the game by zero and by multiplying all entries above the diagonal with negative one. It turns out that the existence of a particular strict improvement cycle corresponds precisely to the property that the modified Jacobian is semipositive (Fiedler and Pták 1966). 6 We transform these conditions in several steps, using a powerful recursive characterization of semipositivity due to Johnson et al. (1994), as well as the technique of flipping around individual strategy spaces (Vives 1990;Amir 1996). These steps lead us ultimately to the local analogue of the Monderer-Shapley condition.
Finally, we apply our necessary conditions to two selected classes of games that traditionally have been studied by economists. These are the lottery contest with heterogeneous valuations, and a differentiated Bertrand game with linear demand and quadratic costs. For these classes of games, our necessary conditions turn out to be also sufficient for the existence of a generalized ordinal potential.
Related literature. Complete characterizations of exact and generalized ordinal potential games are known for the class of finite games (Monderer and Shapley 1996a;Milchtaich 1996). For the class of strategic games, Monderer and Shapley (1996a) provided a complete characterization of exact potential games. Voorneveld and Norde (1997) have shown that a strategic game admits an ordinal potential if and only if there are no weak improvement cycles and an order condition is satisfied. 7 Exact potential games admit a convenient characterization in the class of smooth games, i.e., in the class of games with interval strategy spaces and twice continuously differentiable payoff functions. Specifically, a smooth game admits an exact potential if and only if the Jacobian of that game, i.e., the matrix of cross-derivatives of players' payoff functions, is globally symmetric (Monderer and Shapley 1996a). This characterization extends in a straightforward way to weighted potential games. However, no differentiable characterization has been available up to this point for the classes of ordinal potential or generalized ordinal potential games. 8 6 The relevant elements of the theory of semipositive matrices will be reviewed below. 7 For a rigorous statement of this important result, we refer the reader to Voorneveld and Norde (1997). 8 Monderer and Shapley (1996a, p. 135) wrote: "Unlike (weighted) potential games, ordinal potential games are not easily characterized. We do not know of any useful characterization, analogous to the one given in (4.1), for differentiable ordinal potential games." Since then, the problem has apparently remained unaddressed. See, e.g., the recent surveys by Mallozzi (2013), González-Sánchez and Hernández-Lerma (2016), or Lã et al. (2016).

3
The rest of the paper is structured as follows. Section 2 contains preliminaries. Section 3 discusses strict improvement cycles involving two players. The case of more than two involved players is dealt with in Sect. 4. Section 5 presents the local analogue of the Monderer-Shapley condition. Section 6 discusses boundary equilibria and related issues. Applications are provided in Sect. 7. Section 8 concludes. All the proofs have been relegated to an Appendix.

Preliminaries
Let be a (non-cooperative) n-player game with set of players N = {1, … , n} , strategy space X i for each player i ∈ N , and payoff function u i ∶ X N = X 1 × ⋯ × X n → ℝ for each i ∈ N . Given a strategy profile x N ≡ (x 1 , … , x n ) ∈ X N , we denote by The game will be called smooth (Vives 1999) and u i is twice continuously differentiable on X N . Further, to denote the interior of the interval X i , we will use the notation X i = (x i , x i ). 9 Following Monderer and Shapley (1996a), a non-cooperative game is called an ordinal potential game [a generalized ordinal potential game] if there exists a function P ∶ X N → ℝ such that for any i ∈ N , x i ∈ X i , x i ∈ X i , and x −i ∈ X −i . Note that no continuity nor differentiability assumptions are imposed on the (generalized) ordinal potential functions. 10 Given an n-player game , a strict improvement cycle of length L is a finite nondegenerate sequence of strategy profiles in X N with the property that, for any l = 0, … , L − 1 , there is a player i = (l) ∈ N such that x l+1 , where x L N should be read as x 0 N . As pointed out by Voorneveld and Norde (1997), the existence of an ordinal potential in a given game is, under some additional condition, equivalent to the nonexistence of a weak improvement cycle (wherein payoff has to increase strictly for Thus, we restrict attention to one-dimensional strategy spaces. The extension to multi-dimensional strategy spaces is discussed in the conclusion. 10 This point is worth being mentioned because, as pointed out by Voorneveld (1997), a continuous ordinal potential game need not, in general, admit a continuous ordinal potential. See also Peleg et al. (1996).

3
Ordinal potentials in smooth games some path transition only, and weakly otherwise). In the present analysis, we will need only the following necessary condition.
Lemma 1 (Voorneveld 1997) Let be a generalized ordinal potential game. Then, there is no strict improvement cycle in .

Strict improvement cycles involving two players
We start the analysis by considering local strict improvement cycles that involve precisely two players. For this case, the approach outlined in the introduction leads to the following observation.
Proposition 1 Suppose that the smooth n-player game admits a generalized ordinal potential. Then, for any two players i, j ∈ N with i ≠ j , it holds that Thus, if a smooth n-player game admits a generalized ordinal potential, then at any strategy profile at which marginal payoffs vanish in the interior for two players, the product of the corresponding cross-derivatives is nonnegative. In particular, any smooth generalized ordinal potential game necessarily exhibits pairwise strategic substitutes or complements at any interior Cournot-Nash equilibrium. 11 Proposition 1 implies that games in which one player's best-response function is strictly increasing and another player's best-response function is strictly decreasing is never a generalized ordinal potential game. An example is the mixed oligopoly model by Singh and Vives (1984), in which one firm chooses price, and the other firm chooses quantity. Another instance is quantity competition between a dominant firm and several fringe firms (Bulow et al. 1985). Likewise, asymmetric contests (Dixit 1987) never admit a generalized ordinal potential. Many similar examples, taken from diverse areas such as law enforcement, business strategy, and citizen protests, can be found in Tombak (2006), Monaco and Sabarwal (2016), and Barthel and Hoffmann (2019), for instance.
In games with more than two players, condition (3) requires pairwise strategic complements or substitutes only. Therefore, unless sum-aggregative, 12 an ordinal potential game need not exhibit either strategic complements or strategic substitutes at x N . Indeed, flipping around the natural order of the strategy space of one of three players, say, may certainly destroy the property of strategic complements or strategic substitutes, but it does not affect the pairwise property, nor does it affect the property of being an ordinal potential game.
To understand why Proposition 1 holds true, consider Fig. 1. Keeping the strat- is assumed to be well-defined and strictly increasing in player j's strategy x j , while player j's best-response function in conflict with the necessary condition. And indeed, for > 0 small enough, the circular path starting at the upper left corner of the depicted square and running clockwise around it, i.e., the finite sequence constitutes a strict improvement cycle, as will be explained now. To start with, consider the strategy change corresponding to the upper side of the square. Then, with Constructing a strict improvement cycle involving two players 13 As usual, provided that 2 u i ∕ x 2 i < 0 holds globally, player i's marginal condition defines a function holds in the interior. We will refer to i as player i's best-response function. In particular, for any other player j ≠ i , we may refer to as the slope of player i's best-response function with respect to player j (in the interior). Thus, ij (x N ) ≥ 0 ( ≤ 0 ) if and only if exhibits strategic complements (strategic substitutes) between i and j at x N .
Ordinal potentials in smooth games small, player i's payoff is first increasing (over a longer section of the side) and then decreasing (over a shorter section of the side). The point to note is that, as a consequence of smoothness of payoffs at the Cournot-Nash equilibrium, player i's payoff function along the upper side of the square may be approximated arbitrarily well by a parabola opening downwards, provided the square is small enough. As the parabola is symmetric around its peak, the payoff difference for player i, when switching from strategy x i − to x i + , will be overall positive. 14 Similar considerations apply to the remaining three sides of the square. In fact, at the bottom side, there is no trade-off because player i's marginal payoff is always negative there. Thus, in sum, one may construct a strict improvement cycle that leads around the equilibrium. As noted in the previous section, however, this is incompatible with the existence of a generalized ordinal potential.

Strict improvement cycles involving more than two players
In this section, we consider strict improvement cycles that involve more than two players. We will initially restrict attention to a narrow class of paths in which the players involved consecutively raise their respective strategies, taking turns in their natural order, and subsequently lower their strategies, following the same order. Figure 2 illustrates a path of this kind for the case of three players. 15 It turns out that the local feasibility of a strict improvement cycle is, for general n ≥ 2 , determined by the semipositivity (Fiedler and Pták 1966;Johnson et al. 1994) of a modified matrix that is derived from the Jacobian of the game by changing the sign of all entries above the diagonal, and by replacing all entries on the diagonal by zero. Fig. 2 Constructing a strict improvement cycle involving three players 14 There is a minor technical subtlety here in so far that the payoff difference approaches zero as goes to zero. However, as shown in the Appendix with the help of a careful limit consideration, the payoff difference approaches zero from above since the corresponding cross-derivative is positive. This turns out to be sufficient to settle the trade-off for a sufficiently small but still positive . 15 In contrast to the previous section, we will allow for edges that are not necessarily of equal length.

3
Formally, let N = ( 1 , … , n ) T ∈ ℝ n be a vector, where the T indicates transposition, as usual. We will write N > 0 if all entries of N are positive, i.e., if i > 0 for all i = 1, … , n.
Definition 1 (Fiedler and Pták 1966) Semipositivity generalizes the concept of a P-matrix (Gale and Nikaidô 1965). Intuitively, the fact that A is semipositive says that the interior of the convex cone generated by the column vectors of A intersects the positive orthant ℝ n ++ = {z N ∈ ℝ n ∶ z N > 0} . Thus, checking semipositivity of a matrix corresponds to solving a feasibility problem in linear programming. Moreover, semipositivity is a robust property, and one may think of it in the present context as a generalized sign test on matrix entries.
An analysis of the conditions for the described path to constitute a strict improvement cycle leads to the following observation.
Proposition 2 Suppose that the smooth n-player game admits a generalized ordinal potential. Then, at any strategy profile x N ∈ X N such that the modified Jacobian cannot be semipositive.
This result may be summarized as follows. Consider a smooth generalized ordinal potential game and take any interior strategy profile x N at which marginal payoffs vanish for all players. For instance, x N could be an interior Cournot-Nash equilibrium. Then, the modified Jacobian at x N must not be semipositive.
There are two basic ways to generate even more stringent necessary conditions. First, one may permute the order in which players change their respective strategies. Second, one may flip around the natural order of individual strategy spaces. The general treatment of these extensions, accomplished in the working paper version (Ewerhart 2017, Sect. 4.2), turns out to be obsolete for the purpose of the present x n x n−1 2 u n (x N ) Ordinal potentials in smooth games analysis. However, we will use instances of permutations and flipped strategy spaces in the proof of our main result. 16

A local analogue of the Monderer-Shapley condition
From the necessary conditions identified in the previous sections, one may derive the following theorem, which is the main result of the present paper.
Theorem 1 Consider an interior strategy profile x * N in a smooth n-player game such that and such that all cross-derivatives admits a generalized ordinal potential, then there exist positive weights Thus, as outlined in the Introduction, the existence of an ordinal potential implies a local property that is analogous to the global differentiable condition for a weighted potential game. To see the theorem at work, assume that payoffs admit the representation where X i ⊆ ℝ ++ for all i ∈ N , and ∶ X N → ℝ is an arbitrary twice continuously differentiable function that does not depend on i. Then, as pointed out by Kukushkin (1994), the mapping is an ordinal potential. 17 Suppose that x * N is an interior Cournot-Nash equilibrium. Then, Theorem 1 implies the existence of weights (10) holds. This is indeed the case for For a recent application, see Nocke and Schutz (2018). 16 As will be explained in Sect. 6, no additional conditions can be obtained by considering strategy profiles at which less than two first-order conditions hold. Similarly, allowing for more complicated local paths does not tighten our conditions. For a discussion of this point, see likewise the working paper version (Ewerhart 2017).
as can be checked in a straightforward way. 18 Theorem 1 assumes that all cross-derivatives are non-vanishing. As the following example shows, this assumption cannot be dropped without losing the conclusion of Theorem 1.
This game is easily seen to admit the ordinal potential P(x 1 , x 2 ) = u 1 (x 1 , x 2 ) . However, at the interior Cournot-Nash equilibrium x * N = (0, 0) , the cross-derivatives are given by in conflict with relationship (10).
Thus, intuitively, the assumption that cross-derivatives do not vanish is needed because strictly monotone transformations of a player's payoff, under which the class of ordinal potential games is invariant, may possess points with zero slope.
The proof of Theorem 1 has three main steps. First, one considers a variety of strict improvement cycles involving either two or three players to show that, at any profile x * N , and for any set {i, j, k} ⊆ N of pairwise different players, Then, an induction argument is used to obtain an analogous result for more than three players. In a final step, the weight for player i is defined as 18 Making use of player j's first-order condition we see that is symmetric with respect i and j. Thus, (10) indeed holds for the suggested local weights. x Ordinal potentials in smooth games and the consistency relations derived in the previous two steps are used to show that these weights have the desired property (10).

Boundary equilibria and related issues 19
While stated for interior strategy profiles at which all players have zero marginal incentives, it should be clear that Theorem 1 has immediate implications also for many strategy profiles x * N that do not satisfy these conditions. Indeed, it suffices to ignore any player whose strategy choice either exhibits strict marginal incentives or lies at the boundary (or both), and to consider the resulting game with less than n players. This trick, however, does not work for every strategy profile. Specifically, Theorem 1 does not deliver a local necessary condition at strategy profiles where either (1) at most one player's marginal incentives are zero, or (2) even though there are two or more players whose marginal incentives are zero, at most one of those chooses an interior strategy. In this section, we will discuss these cases.
We start by looking at the first case outlined above, i.e., we consider a strategy profile x * N at which at most one player's marginal incentives are zero. It turns out that, under such condition, there cannot be a strict improvement cycle locally at x * N . To the contrary, as the following result shows, it is then always feasible to locally construct an ordinal potential.

Proposition 3 Let x *
N ∈ X N be a strategy profile (possibly but not necessarily located at the boundary) in a smooth n-player game , and let i ∈ N be a player such that 20 Then, admits an ordinal potential in a small neighborhood of x * N .
Thus, quite generally, strategy profiles that do not entail zero marginal incentives for at least two players cannot be used to identify local necessary conditions for the existence of an ordinal or generalized ordinal potential. In particular, this is so for equilibria in which at most one player chooses an interior strategy while all others choose a boundary strategy under positive shadow costs. (20) The proof of Proposition 1 is based on a simple idea. Condition (21) implies that the payoff functions of all players except possibly player i are locally strictly monotone at x * N . Therefore, a natural candidate for an ordinal potential is a function P(x i , x −i ) that adds together player i's payoff function u i (x i , x −i ) and a suitable linear function in the vector x −i . Indeed, it is not hard to see that, provided that the slope of the linear function with respect to x j , for any j ≠ i , is chosen sufficiently large in absolute terms so as to dominate the corresponding slope of u i (x i , x −i ) with respect to x j , the function P(x i , x −i ) becomes an ordinal potential for in a small neighborhood of x * N . The situation differs in the second case outlined above, i.e., when there are at least two players that have zero marginal incentives, but at most one of them chooses an interior strategy. Note that, in this case, a boundary strategy is chosen even though shadow costs vanish. Given the non-robustness of this type of situation, we confine ourselves to the analysis of an illustrative example involving two players, where one player chooses a (lower) boundary strategy and another player chooses an interior strategy. 21 The consideration of a suitable family of strict improvement cycles of the shape outlined in Fig. 3 delivers the following result. 1 3

Ordinal potentials in smooth games
Thus, in the case of a non-robust boundary equilibrium (and in similar cases), we obtain an additional local necessary condition, but this condition is somewhat weaker than the one obtained for interior equilibria.
To understand this result, note that for the player that chooses an interior strategy, the sign of the payoff differential along the boundary segment shown in Fig. 3 will be determined by the third (or even higher) derivative of her payoff function. Furthermore, while the proofs of Propositions 3 and 4 differ, the two results are intuitively closely related. Specifically, the third-order condition in Proposition 4 captures a local monotonicity condition that needs to be satisfied to disallow the local construction of an ordinal potential as done in the proof of Proposition 3.

Applications
In this section, we review two classes of games traditionally studied by economists, and observe that, for these games, our necessary conditions are also sufficient. 22

Contests
In the n-player lottery contest with valuations V 1 > 0, … , V n > 0 , player i's payoff is given by where we assume that X 1 = ⋯ = X n = [0, V max ] , with V max = max i∈N V i . It follows from a general result of Szidarovszky and Okuguchi (1997) that this game has a unique (yet not necessarily interior) Cournot-Nash equilibrium x * N = (x * 1 , … , x * n ). Rather than applying our criterion to the n-player equilibrium, we will consider an equilibrium in the two-player game between arbitrary players i, j ∈ N with j ≠ i , assuming that all remaining players remain inactive. So let be such that the following conditions hold: In the bilateral game between players i and j, equilibrium efforts are given by the well-known expressions (cf. Konrad 2009) The cross-derivative for player i is given by An analogous expression may be derived for player j. We therefore see that the necessary condition stated as Proposition 1 holds if and only if V i = V j . On the other hand, if V 1 = ⋯ = V n , then the lottery contest can be shown to admit a generalized ordinal potential (Ewerhart and Kukushkin 2019). Thus, in the case of the lottery contest with heterogeneous valuations, our criterion is indeed not only necessary but also sufficient. 23

Differentiated Bertrand competition
There are n ≥ 2 firms i = 1, … , n with differentiated products. Suppose that each firm i ∈ N chooses a price p i ≥ 0 , and subsequently sells a quantity where Q i > 0 , s i > 0 , and ij ≠ 0 are arbitrary parameters. 24 Firm i's cost function is given by C i (q i ) = c i q 2 i , where c i > 0 . Thus, firm i's profit reads Suppose that there exists a price vector p * N > 0 at which marginal profits vanish (and at which profits are positive), for all firms. 25 Cross-derivatives are given by 23 This example illustrates likewise the local nature of our conditions. Indeed, for x i ≠ x j , the product of the cross-derivatives is negative in the lottery contest. 24 If ij = 0 for some i ≠ j , then the necessary and sufficient conditions (31, 32) identified below must be complemented by additional conditions analogous to (32) for any number m ∈ {4, … , n} of players. Moreover, the functional form of the potential may differ somewhat from (33). In a nutshell, one first notes that, for a generalized ordinal potential to exist in the Bertrand game, it is necessary that, for any i, j ∈ N with i ≠ j , it holds that ij = 0 if and only if ji = 0 . Then, in any connected component of the directed weighted graph defined by the bilateral price externalities { ij } , one chooses a spanning tree and defines a potential weight for a firm (or node) analogous to (35) by taking the product over all absolute price externalities represented as weights in the tree directed toward the firm. The details are omitted. 25 This is the case, for instance, if max j≠i | | | ij | | | is sufficiently small.

Ordinal potentials in smooth games
Note that 1 + 2s i c i > 0 for i ∈ N . Therefore, from Theorem 1, necessary for the existence of a generalized ordinal potential is that 26 and Conversely, we have the following observation. (31) and (32) hold. Then, with is a generalized ordinal potential for the differentiated Bertrand game.

Proposition 5 Suppose that conditions
Thus, also in this class of games, our conditions turn out to both necessary and sufficient for the existence of a generalized ordinal potential. 27

Concluding remarks
In this paper, we have identified conditions necessary for the existence of a generalized ordinal potential in any game with interval strategy spaces and twice continuously differentiable payoff functions. For selected classes of games, including lottery contests with heterogeneous valuations, and differentiated Bertrand games with linear demand and quadratic costs, these conditions are also sufficient. Thus, a first step toward the differentiable characterization of ordinal and generalized ordinal potential games has been made.
Further extensions are possible. For example, using similar methods, a necessary differentiable condition may be obtained from the analysis of non-local strict improvement cycles. This idea is developed in the working paper version (Ewerhart 2017). There as well, we extend Propositions 1 and 2 to games with multi-dimensional strategy spaces. However, given the lack of a suitable generalization of the notion of semipositivity, it has to remain an open question if the local analogue of the Monderer-Shapley condition extends accordingly. 28 Echenique (2004) has shown that, generically, a finite two-player ordinal potential game is a game of strategic complements (potentially after reordering strategic spaces), yet an ordinal potential game with more than two players need not generically be a game of strategic complements. While it is not straightforward to compare the two settings, some intuitions carry over. In particular, our findings provoke the question if any finite ordinal potential game with more than two players can be turned, by reordering of strategy spaces, into a game of pairwise strategic complements and substitutes. We do not know the answer to this question.
Any smooth exact potential game with negative definite Hessian is a concave game in the sense of Rosen (1965). 29 The converse, however, is not true in general. In fact, our results imply that a concave game need not even admit a generalized ordinal potential. Indeed, the lottery contest is concave (Ewerhart and Quartieri 2019), but as seen above, it does not admit a generalized ordinal potential unless valuations are homogeneous.
There are three natural classes of smooth games that satisfy our most stringent necessary conditions. These are symmetric two-player zero-sum games, supermodular sum-aggregative game, and symmetric games that admit only symmetric equilibria. 30 To the extent that necessary conditions are indicative of sufficient conditions, one might speculate that these classes of games, potentially under additional technical conditions, have the property that an ordinal potential can be constructed locally. 31 Acknowledgements The idea for this work originated during an inspiring conversation with Nikolai Kukushkin. Two anonymous referees provided valuable feedback on an earlier version. Material contained herein has been presented at the Department of Business of the University of Zurich and at the National University of Singapore. For useful discussions, I am grateful to Rabah Amir, Georgy Egorov, Josef Hofbauer, Felix Kübler, Stephen Morris, Georg Nöldeke, Marek Pycia, Karl Schmedders, and Satoru Takahashi.

Appendix: Proofs
Proof of Proposition 1 By contradiction. Suppose that, at some profile x * N ∈ X N , and for some players i and j with i ≠ j , we have 28 A multi-dimensional variant of the Monderer-Shapley condition for exact potential games can be found in Deb (2008Deb ( , 2009. 29 See Neyman (1997), Ui (2008), and Hofbauer and Sandholm (2009). 30 Smooth symmetric games admit at least one symmetric equilibrium under standard assumptions (Moulin 1986, p. 115). However, there are also large classes of economically relevant symmetric games that admit only asymmetric pure-strategy Nash equilibria (cf. Amir et al. 2010). 31 The author is presently exploring the validity of these conjectures.

3
Ordinal potentials in smooth games By renaming players, if necessary, we may assume w.l.o.g. that which corresponds to the case shown in Fig. 1. It is claimed now that, for any sufficiently small > 0 , the payoff difference corresponding to the upper side of the square satisfies To prove this, we determine the second-order Taylor approximation of + i ( ) at = 0 . Writing f ( ) for + i ( ) , our differentiability assumptions combined with Taylor's theorem imply that there is a remainder term r(.) with lim →0 r( ) = 0 such that for any sufficiently small > 0, Clearly, f (0) = 0 . As for the first derivative f � (0) , one obtains Evaluating at = 0 , and subsequently exploiting the necessary first-order condition for player i at the interior equilibrium x * N , we find Next, consider the second derivative of + i ( ) at = 0 , i.e., Invoking Schwarz's theorem regarding the equality of cross-derivatives for twice continuously differentiable functions, and subsequently using (38), one obtains In sum, we have shown that f � (0) = f (0) = 0 and f �� (0) > 0 . Thus, using (40), it follows that + i ( ) > 0 for any sufficiently small > 0 . Analogous arguments can be used to deal with the other three sides of the square. Specifically, one defines and now readily verifies that and that

3
Ordinal potentials in smooth games It follows that + i ( ) > 0 , + j ( ) > 0 , − i ( ) > 0 , and − j ( ) > 0 all hold for > 0 small enough. But then, the finite sequence (6) is a strict improvement cycle, which is incompatible with the existence of a generalized ordinal potential by Lemma 1. □

Proof of Proposition 2
Suppose that the modified Jacobian J = J(x N ) is semipositive. Then, by definition, there exists a vector N = ( 1 , … , n ) T ∈ ℝ n with N > 0 such that J N > 0 . Consider the finite sequence where > 0 is a small constant as before. Figure 2 illustrates this path for n = 3 , where the rectangular-shaped box has sides of respective length i = i for i = 1, 2, 3 . It is claimed that, for any > 0 sufficiently small, the following four conditions hold: To establish claim (1), we proceed as in the proof of Proposition 1, and consider the first two derivatives of the payoff difference at = 0 . The first derivative of (1,+) ( ) at = 0 is given by Hence, from player 1's first-order condition, Next, one considers the second derivative of (1,+) ( ) at = 0 , i.e., Collecting terms, one obtains Thus, using 1 > 0 , and noting that the expression in the curly brackets corresponds to the first component of the vector J N > 0 , one arrives at It follows that (1,+) ( ) > 0 for any > 0 sufficiently small, which proves claim (1). To verify claims (2)  In sum, this proves claims (2) through (4). Thus, there exists a strict improvement cycle in the generalized ordinal potential game . Since this is impossible, the proposition follows. □ The three lemmas below will be used in the proof of Theorem 1. Following the literature, we will call a square matrix A ∈ ℝ n×n inverse nonnegative if the matrix (67) (i,+) (0) = 0 (i = 2, … , n), 1 3 inverse A −1 exists and if, in addition, all entries of A −1 are nonnegative. The following lemma provides a useful recursive characterization of semipositivity.
Lemma 2 (Johnson et al. 1994 Consider now a small circular path along the edges of a small three-dimensional rectangular-shaped box around x * N . Along the path, players i, j, and k move in this order, with i and k initially increasing their strategies, while j initially decreases her strategy. Since this corresponds to flipping around player j's strategy space, all 1 3 2. Weak strategic substitutes at x * N between player i and each of players j and k, as well as weak strategic complements at x * N between players j and k. In either case, by flipping around the strategy space of player i, the game may be transformed into a game that exhibits either weak strategic substitutes at x * N or weak strategic complements at x * N . Since the operation of flipping around individual strategy spaces does not affect the validity of Eq. (77), we find that the conclusion indeed holds in the general case. □ Lemma 4 Suppose that the smooth n-player game admits a generalized ordinal potential. Then, at any interior strategy profile x * N at which all first-order conditions hold, and for any set of pairwise distinct players {i 1 , … , i m } ⊆ N with m ≥ 3, using the notation introduced in (76), it holds that Proof The proof proceeds by induction. The case m = 3 follows directly from Lemma 3. Suppose that m ≥ 4 , and let {i 1 , i 2 , … , i m } be an arbitrary set of pairwise different players. Suppose that the claim has been shown for any m � ∈ {3, 4, … , m − 1} . Then, a consideration of the two subsets {i 1 , i 2 , … , i m−1 } and Combining the two equations via multiplication yields By assumption, i 1 i m−1 ⋅ i m−1 i 1 ≠ 0 . Hence, eliminating these common nonzero factors, (88) implies as claimed. This concludes the induction step, and therefore proves the lemma. □ Proof of Theorem 1 Let x * N be an interior strategy profile such that all first-order conditions hold at x * N and such that ij ≠ 0 for all i ≠ j . We need to find positive constants w 1 > 0, … , w n > 0 such that It is claimed that ij w i = ji w j (i, j ∈ N, j ≠ i).
x j (x j −x j ).

Proof of Proposition 5
It will be shown that P is a weighted potential. Since any weighted potential is, in particular, a generalized ordinal potential, this is sufficient to prove the proposition. So consider some player i ∈ N , prices p ′ i ≥ 0 and p ′′ i ≥ 0 , as well as a price vector p −i ∈ ℝ n−1 + . It is claimed that To see this, note first that But this was shown in the proof of Theorem 1. Since � w i > 0 for i ∈ N , we conclude that P is indeed a weighted potential for the differentiated Bertrand game. The proposition follows. □