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, 1996a). However, no analogous characterization is known for ordinal potential games. The present paper derives simple 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 if a speci…c game admits a potential can be quite valuable. For example, the existence of a potential reduces the problem of …nding a Nash equilibrium to a straightforward optimization problem. Therefore, it is of some interest to know the conditions under which a potential exists. Clearly, su¢ cient 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 su¢ cient 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 payo¤ functions are twice continuously di¤erentiable (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 …nite circular sequence of strategy pro…les with the property that moving to the next pro…le in the sequence amounts to one player strictly raising her payo¤ by a unilateral change in strategy. However, in general, identifying strict improvement cycles in smooth games is not straightforward. We therefore consider a speci…c path in the neigh-1 Both exact and ordinal concepts have been considered in the literature. For a function P on the space of strategy pro…les to be an exact potential (a weighted potential), the di¤erence in a player's payo¤ resulting from a unilateral change of her strategy must precisely (up to a positive factor) equal the corresponding di¤erence in P . For a function to be an ordinal potential (a generalized ordinal potential), any weak or strict gain (any strict gain) in a player's payo¤ resulting from a unilateral change of its strategy must be re ‡ected by a corresponding gain in P . 2 Applications of potential methods are vast and include, for example, the analysis of oligopolistic markets (Slade, 1994), learning processes (Monderer and Shapley, 1996b;Fudenberg and Levine, 1998;Young, 2004), population dynamics (Sandholm, 2001(Sandholm, , 2009Cheung, 2014), the robustness of equilibria (Frankel et al., 2003;Morris and Ui, 2005;Okada and Tercieux, 2012), the decomposition of games (Candogan et al., 2011), imitation strategies (Duersch et al., 2012), dynamics (Candogan et al., 2013a(Candogan et al., , 2013b, equilibrium existence (Voorneveld, 1997;Kukushkin, 1994Kukushkin, , 2011, solution concepts (Peleg et al., 1996;Tercieux and Voorneveld, 2010), games with monotone best-response selections (Huang, 2002;Dubey et al., 2006;Jensen, 2010), supermodular and zero-sum games (Brânzei et al., 2003), and even mechanism design (Jehiel et al., 2008). 3 Since any ordinal potential game is, in particular, a generalized ordinal potential game, we also obtain necessary conditions for the existence of an ordinal potential.
borhood of a …xed strategy pro…le x N , and shrink it to virtually in…nitesimal size, as if using a pantograph. 4 This di¤erential 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 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 pro…le at which at least two …rst-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 …rst 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 …xed strategy pro…le x N at its center. To formalize the resulting condition on cross-derivatives, we introduce a modi…ed 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 modi…ed 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 ‡ipping around individual strategy spaces (Vives, 1990;Amir, 1996). These steps lead us ultimately to the local analogue of the  A pantograph is a mechanical drawing instrument that allows creating copies of a plan on a di¤erent scale. 5 To caution the reader, we stress that, despite a similarity in terminology, the present analysis is not directly related to the use of local potentials in the analysis of informational robustness (Morris, 1999;Frankel et al., 2003;Morris and Ui, 2005;Okada and Tercieux, 2012). 6 The relevant elements of the theory of semipositive matrices will be reviewed below.
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 di¤erentiated Bertrand game with linear demand and quadratic costs. For these classes of games, our necessary conditions turn out to be also su¢ cient 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 …nite 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 there are no weak improvement cycles and an order condition is satis…ed. 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 di¤erentiable payo¤ functions. Speci…cally, 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' payo¤ functions, is globally symmetric (Monderer and Shapley, 1996a). This characterization extends in a straightforward way to weighted potential games. However, no di¤erentiable characterization has been available up to this point for the classes of ordinal potential or generalized ordinal potential games. 8 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 Section 4. Section 5 presents the local analogue of the Monderer-Shapley condition. Section 6 discusses boundary equilibria and related issues. Applications are provided in Section 7. Section 8 concludes. All the proofs have been relegated to an Appendix.
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 di¤erentiable 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).

Preliminaries
Let be a (noncooperative) n-player game with set of players N = f1; :::; ng, strategy space X i for each player i 2 N , and payo¤ function u i : X N = X 1 ::: X n ! R for each i 2 N .
Given a strategy pro…le x N (x 1 ; :::; x n ) 2 X N , we denote by x i = (x 1 ; :::; x i 1 ; x i+1 ; :::; x n ) the pro…le composed of the strategies chosen by the opponents of player i, so that x i 2 X i = X 1 ::: X i 1 X i+1 ::: X n . Thus, u i (x N ) = u i (x i ; x i ), etc. The game will be called smooth (Vives, 1999) if for any i 2 N , the set X i is a non-degenerate compact interval 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 noncooperative game is called an ordinal potential game [a generalized ordinal potential game] if there exists a function P : for any i 2 N , x i 2 X i , b x i 2 X i , and x i 2 X i . Note that no continuity nor di¤erentiability assumptions are imposed on the (generalized) ordinal potential functions. 10 Given an n-player game , a strict improvement cycle of length L is a …nite nondegenerate sequence of strategy pro…les ::: ! x 0 N ! x 1 N ! ::: ! x L 1 N ! ::: in X N with the property that, for any l = 0; :::; L 1, there is a player i = (l) 2 N such that 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 non-existence of a weak 9 Thus, we restrict attention to one-dimensional strategy spaces. The extension to multidimensional 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). improvement cycle (wherein payo¤ has to increase strictly for 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 2 N with i 6 = j, it holds that Thus, if a smooth n-player game admits a generalized ordinal potential, then at any strategy pro…le at which marginal payo¤s 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 …rm chooses price, and the other …rm chooses quantity. Another instance is quantity competition between a dominant …rm and several fringe …rms (Bulow et al., 1985).
Likewise, asymmetric contests (Dixit, 1987) never admit a generalized ordinal potential. Many 11 Similar local necessary conditions may be obtained from the consideration of boundary equilibria, as will be explained in Section 6. 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 Ho¤mann (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, ‡ipping 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 a¤ect the pairwise property, nor does it a¤ect the property of being an ordinal potential game. To understand why Proposition 1 holds true, consider Figure 1. Keeping the strategy pro…le x i;j = (x 1 ; :::; x i 1 ; x i+1 ; :::; x j 1 ; x j+1 ; :::; x n ) …xed, player i's best-response function 13 i = 12 See, e.g., Corchón (1994). 13 As usual, provided that @ 2 u i =@x 2 i < 0 holds globally, player i's marginal condition de…nes a function i = i (x i ) that maps any vector x i 2 X i to a unique i (x i ) 2 X i such that @u i ( i (x i ); x i )=@x i = 0 holds in the interior. We will refer to i as player i's best-response function. In particular, for any other player j 6 = 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 .
i (x j ; x i;j ) is assumed to be well-de…ned and strictly increasing in player j's strategy x j , while player j's best-response function j = j (x i ; x i;j ) is strictly decreasing in player i's strategy x i . Therefore, in con ‡ict 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 …nite 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 " small, player i's payo¤ is …rst 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 payo¤s at the Cournot-Nash equilibrium, player i's payo¤ 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 payo¤ di¤erence 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-o¤ because player i's marginal payo¤ 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.
14 There is a minor technical subtlety here in so far that the payo¤ di¤erence approaches zero as " goes to zero. However, as shown in the Appendix with the help of a careful limit consideration, the payo¤ di¤erence approaches zero from above since the corresponding cross-derivative is positive. This turns out to be su¢ cient to settle the trade-o¤ for a su¢ ciently small but still positive ".

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 semi-positivity (Fiedler and Pták, 1966;Johnson et al., 1994) of a modi…ed 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. Formally, let N = ( 1 ; :::; n ) T 2 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.
De…nition 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 15 In contrast to the previous section, we will allow for edges that are not necessarily of equal length. columns of A intersects the positive orthant R n ++ = fz N 2 R n : z N > 0g. 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 pro…le x N 2 X N such that x 1 2 X 1 ; :::; x n 2 X n , and @u 1 (x N ) @x 1 = :: the modi…ed Jacobian This result may be summarized as follows. Consider a smooth generalized ordinal potential game, and take any interior strategy pro…le x N at which marginal payo¤s vanish for all players.
For instance, x N could be an interior Cournot-Nash equilibrium. Then, the modi…ed 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 ‡ip around the natural order of individual strategy spaces. The general treatment of these extensions, accomplished in the working paper version (Ewerhart, 2017, Sec. 4.2), turns out to be obsolete for the purpose of the present analysis. However, we will use simple instances of permutations and ‡ipped strategy spaces in the proof of our main result. 16

A local analogue of the Monderer-Shapley condition
From the necessary conditions identi…ed 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 pro…le x N in a smooth n-player game such that and such that all cross-derivatives If admits a generalized ordinal potential, then there exist positive weights w 1 (x N ) > 0; :: Thus, as outlined in the Introduction, the existence of an ordinal potential implies a local property that is analogous to the global di¤erentiable condition for a weighted potential game.
To see the theorem at work, assume that payo¤s admit the representation where X i R ++ for all i 2 N , and : X N ! R is an arbitrary twice continuously di¤erentiable 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 w 1 (x N ) > 0; :::; w n (x N ) > 0 such that (10) holds. 16 As will be explained in Section 6, no additional conditions can be obtained by considering strategy pro…les at which less than two …rst-order conditions hold. Similarly, allowing for more complicated local paths does not tighten our conditions. For discussion of this point, see the working paper version (Ewerhart, 2017). 17 For a recent application, see Nocke and Schutz (2018). This is indeed the case for 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 con ‡ict with relationship (10).
Thus, intuitively, the assumption that cross-derivatives do not vanish is needed because strictly monotone transformations of a player's payo¤, 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 pro…le x N , and 18 Making use of player j's …rst-order condition we see that is symmetric with respect i and j. Thus, (10) indeed holds for the suggested local weights.
for any set fi; j; kg N of pairwise di¤erent players, Then, an induction argument is used to obtain an analogous result for more than three players.
In a …nal step, the weight for player i is de…ned as and the consistency relations derived in the previous two steps are used to show that these weights have the predicted property (10).

Boundary equilibria and related issues 19
While stated for interior strategy pro…les at which all players have zero marginal incentives, it should be clear that Theorem 1 has immediate implications also for many strategy pro…les x N that do not satisfy these conditions. Indeed, it su¢ ces 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 pro…le. Speci…cally, Theorem 1 does not deliver a local necessary condition at strategy pro…les where either (i) at most one player's marginal incentives are zero, or (ii) 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 …rst look at the …rst case outlined above, i.e., we consider a strategy pro…le 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.
19 I am grateful for an anonymous reviewer for the suggestion to consider boundary equilibria in more detail.
Proposition 3. Let x N 2 X N be a strategy pro…le (possibly but not necessarily located at the boundary) in a smooth n-player game , and let i 2 N be a player such that 20 Then, admits an ordinal potential in a small neighborhood of x N .
Thus, quite generally, strategy pro…les 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.
The proof of Proposition 1 is based on a simple idea. Condition (21) implies that the payo¤ 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 payo¤ 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 6 = i, is chosen su¢ ciently 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 di¤ers 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 con…ne ourselves to the analysis of an illustrative case 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 Figure 3 delivers the following result. 20 As usual, the derivative in (21) is understood to be one-sided if x j = x j or x j = x j . 21 Further generalizations are left for future work.
Proposition 4. Suppose that the smooth n-player game admits a generalized ordinal potential. Then, for any two players i; j 2 N with i 6 = j, 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 payo¤ di¤erential along the boundary segment shown in Figure 3 will be determined by the third (or even higher) derivative of her payo¤ function. Furthermore, while the proofs of Propositions 3 and 4 di¤er, the two results are intuitively closely related. Speci…cally, the third-order condition in Proposition 4 captures a local monotonicity condition that needs to be satis…ed 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 su¢ cient. 22
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 2 N with j 6 = i, assuming that all remaining players remain inactive. So let x # N = (x # i ; x # j ; x # i;j ) 2 X N be such that the following conditions hold: x # i > 0, x # j > 0, x # i;j = (0; :::; 0) 2 R n 2 , and (24) In the bilateral game between players i and j, equilibrium e¤orts 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 summarized in 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 su¢ cient. 23

Di¤erentiated Bertrand competition
There are n 2 …rms i = 1; :::; n with di¤erentiated products. Suppose that each …rm i 2 N chooses a price p i 0, and subsequently sells a quantity where Q i > 0, s i > 0, and ij 6 = 0 are arbitrary parameters. 24 Firm i's cost function is given Suppose that there exists a price vector p N > 0 at which marginal pro…ts vanish (and at which pro…ts are positive), for all …rms. 25 Cross-derivatives are given by Note that 1 + 2s i c i > 0 for i 2 N . Therefore, from Theorem 1, necessary for the existence of a generalized ordinal potential is that 26 23 This example illustrates likewise the local nature of our conditions. Indeed, for x i 6 = x j , the product of the cross-derivatives is negative in the lottery contest. 24 If ij = 0 for some i 6 = j, then the necessary and su¢ cient conditions (31-32) identi…ed below must be complemented by additional conditions analogous to (32) for any number m 2 f4; :::; ng of players. Moreover, the functional form of the potential may di¤er somewhat from (33). In a nutshell, one …rst notes that, for a generalized ordinal potential to exist in the Bertrand game, it is necessary that, for any i; j 2 N with i 6 = j, it holds that ij = 0 if and only if ji = 0. Then, in any connected component in the directed weighted graph de…ned by the bilateral price externalities f ij g, one chooses a spanning tree and de…nes a potential weight for a …rm (or node) analogous to (35) by taking the product over all absolute price externalities represented as weights in the tree directed towards the …rm. The details are omitted. 25 This is the case, for instance, if max j6 =i j ij j is su¢ ciently small. 26 As usual, sgn(:) denotes the sign function, with sgn(z) = 1 if z > 0, sgn(z) = 0 if z = 0, and sgn(z) = 1 if z < 0. and fi; j; k 2 N s.t. i 6 = j 6 = k 6 = ig ) ij jk ki = ji kj ik .
Conversely, we have the following observation.
Thus, also in this class of games, our conditions turn out to both necessary and su¢ cient for the existence of a generalized ordinal potential. 27

Concluding remarks
In this paper, we have identi…ed conditions necessary for the existence of a generalized ordinal potential in any game with interval strategy spaces and twice continuously di¤erentiable payo¤ functions. For selected classes of games, including lottery contests with heterogeneous valuations, and di¤erentiated Bertrand games with linear demand and quadratic costs, these conditions are also su¢ cient. Thus, a …rst step towards the di¤erentiable characterization of ordinal and generalized ordinal potential games has been made.
Further extensions are possible. For example, using similar methods, a necessary di¤erentiable 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 multidimensional 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 …nite 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 …ndings provoke the question if any …nite 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 de…nite 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 su¢ cient 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 28 A multi-dimensional variant of the Monderer-Shapley condition for exact potential games can be found in Deb (2008). 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.

Appendix: Proofs
Proof of Proposition 1. By contradiction. Suppose that, at some pro…le x N 2 X N , and for some players i and j with i 6 = j, we have By renaming players, if necessary, we may assume w.l.o.g. that which corresponds to the case shown in Figure 1. It is claimed now that, for any su¢ ciently small " > 0, the payo¤ di¤erence corresponding to the upper side of the square satis…es To prove this, we determine the second-order Taylor approximation of + i (") at " = 0. Writing f (") for + i ("), our di¤erentiability assumptions combined with Taylor's theorem imply that there is a remainder term r(:) with lim "!0 r(") = 0 such that for any su¢ ciently small " > 0, Clearly, f (0) = 0. As for the …rst derivative f 0 (0), one obtains Evaluating at " = 0, and subsequently exploiting the necessary …rst-order condition for player i at the interior equilibrium x N , we …nd Next, consider the second derivative of + i (") at " = 0, i.e., Invoking Schwarz's theorem regarding the equality of cross-derivatives for twice continuously di¤erentiable functions, and subsequently using (38), one obtains In sum, we have shown that f 0 (0) = f (0) = 0 and f 00 (0) > 0. Thus, using (40), it follows that + i (") > 0 for any su¢ ciently small " > 0. Analogous arguments can be used to deal with the other three sides of the square. Speci…cally, one de…nes and now readily veri…es that and that It follows that + i (") > 0, + j (") > 0, i (") > 0, and j (") > 0 all hold for " > 0 small enough. But then, the …nite 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 modi…ed Jacobian J = J(x N ) is semipositive.
In sum, this proves (ii) through (iv). 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 2 R n n inverse nonnegative if the matrix 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 A.1 (Johnson et al., 1994). A square matrix A 2 R n n is semipositive if and only if at least one of the following two conditions holds: (i) A is inverse nonnegative; (ii) there exists m 2 f1; :::; n 1g and a submatrix b A 2 R n m obtained from A via deletion of n m columns, such that all m m submatrices of b A are semipositive.
Using the lemma above, we may derive the following implication of Proposition 2.
Lemma A.2. Suppose that the smooth n-player game admits a generalized ordinal potential.
Then, at any pro…le x N , and for any set fi; j; kg N of pairwise di¤erent players, Proof. Fix some pro…le x N and a set fi; j; kg N of pairwise di¤erent players such that Using the notation for cross-derivatives, it needs to be shown that ij jk ki = ji kj ik .
Consider now a small circular path along the edges of a small three-dimensional rectangularshaped 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 ‡ipping around player j's strategy space, all cross-derivatives involving player j change sign, so that the corresponding modi…ed Jacobian reads By Proposition 2, J 3 cannot be inverse nonnegative. To prove (77), it su¢ ces to show that the determinant of J 3 , vanishes. To provoke a contradiction, suppose …rst that jJ 3 j > 0. Then, from weak strategic complements at x N , all the entries of the matrix inverse of J 3 , are nonnegative, in contradiction to the fact that J 3 is not inverse nonnegative. Hence, jJ 3 j 0. Suppose next that jJ 3 j < 0. Then, by running through the above-considered path in the opposite direction (i.e., by exchanging the roles of players i and k), Proposition 2 implies is not inverse nonnegative. Clearly, the determinant of b Hence, using an expression for the matrix inverse analogous to (81), all entries of ( b J 3 ) 1 are seen to be nonnegative, in contradiction to the fact that b J 3 is not inverse nonnegative. It follows that jJ 3 j = 0, which proves the claim in the case where exhibits weak strategic complements at x N . The case of weak strategic substitutes, where ij 0, ji 0, ik 0, ki 0, jk 0, and kj 0, is entirely analogous, and therefore omitted. We are now in the position to address the general case. From Proposition 1, we know that exhibits pairwise weak strategic complements or substitutes at x N . Hence, up to a renaming of the players, there are only two remaining cases: (i) Weak strategic complements at x N between player i and each of players j and k, as well as strategic substitutes at x N between players j and k; (ii) 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 ‡ipping 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 ‡ipping around individual strategy spaces does not a¤ect the validity of equation (77), we …nd that the conclusion indeed holds in the general case.
Lemma A.3. Suppose that the smooth n-player game admits a generalized ordinal potential.
Then, at any interior strategy pro…le x N at which all …rst-order conditions hold, and for any set of pairwise distinct players fi 1 ; ::::; i m g N with m 3, using the notation introduced in (76), it holds that provided that i 1 i 3 i 3 i 1 6 = 0; :::; i 1 i m 1 i m i i 1 6 = 0: Proof. The proof proceeds by induction. The case m = 3 follows directly from Lemma A.2.
Suppose that m 4, and let fi 1 ; i 2 ; :::; i m g be an arbitrary set of pairwise di¤erent players.
Suppose that the claim has been shown for any m 0 2 f3; 4; :::; m 1g. Then, a consideration of the two subsets fi 1 ; i 2 ; :::; i m 1 g and fi m 1 ; i m ; i 1 g shows that i 1 i 2 ::: i m 2 i m 1 i m 1 i 1 = i 2 i 1 ::: i m 1 i m 2 i 1 i m 1 , and Combining the two equations via multiplication yields = i 2 i 1 :: By assumption, i 1 i m 1 i m 1 i 1 6 = 0. Hence, eliminating these common nonzero factors, (88) implies i 1 i 2 ::: i m 1 im imi 1 = i 2 i 1 :: as claimed. This concludes the induction step, and therefore proves the lemma.
Proof of Theorem 1. Let x N be an interior strategy pro…le such that all …rst-order conditions hold at x N and such that ij 6 = 0 for all i 6 = j. We need to …nd positive constants w 1 > 0; :::; w n > 0 such that It is claimed that w i = (j 12 j ::: j i 1i j) (j i+1i j ::: j nn 1 j) does the job, with the usual convention that any empty product equals one. Clearly, it su¢ ces to check (90) for i < j, because if i > j, one may just exchange the two sides of equation (90).
Proof of Proposition 3. To construct an ordinal potential locally at x N , start by letting where the sign function sgn(:) is de…ned in Section 7. Next, choose a constant j , for each j 2 N nfig, such that j@u i (x N )=@x j j < j . Then, since is smooth, there is a small rectangular We claim that is an ordinal potential for when joint strategy choices are restricted to U N (x N ). To see this, note …rst that for any x i 2 X i , b x i 2 X i , and x i 2 X i . Next, let j 2 N nfig. Take any x j 2 X j , b x j 2 X j , and x j 2 X j such that (x j ; x j ) 2 U N (x N ) and (b x j ; x j ) 2 U N (x N ). Then, from (102), Invoking the mean value theorem, there exists a strict convex combination e x j of b x j and x j such that Hence, To check the de…nition of an ordinal potential, suppose that u j (x j ; x j ) u j (b x j ; x j ) > 0.
Then, using (101), S j (x j b x j ) > 0. Moreover, since U N (X N ) is convex, j@u i (e x j ; x j )=@x j j < j .
It follows that P (x j ; x j ) P (b x j ; x j ) > 0. Since these steps may be reversed, P is indeed an ordinal potential for in the neighborhood U N (x N ). This proves the proposition.
Proof of Proposition 4. We adapt the proof of Proposition 1. To provoke a contradiction, suppose that There are two cases. Suppose …rst that, as illustrated in Figure 3, Then, condition (109) implies @ 3 u j (x N )=@x 3 j > 0. Given constants i > 0 and j > 0, let b + i (") = u i (x i + i "; x j + j "; x i;j ) u i (x i ; x j + j "; x i;j ), b + j (") = u j (x j j "; x i + i "; x i;j ) u j (x j + j "; x i + i "; x i;j ), b j (") = u j (x j + j "; x i ; x i;j ) u j (x j j "; x i ; x i;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 su¢ cient to prove the proposition. So consider some player i 2 N , prices p 0 i 0 and p 00 i 0, as well as a price vector p i 2 R n 1 + . It is claimed that P (p 0 i ; p i ) P (p 00 i ; p i ) = b w i 1 + 2s i c i (u i (p 0 i ; p i ) u i (p 00 i ; p i )) .
But this was shown in the proof of Theorem 1. Since b w i > 0 for i 2 N , we conclude that P is indeed a weighted potential for the di¤erentiated Bertrand game. The proposition follows.