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Internalization of Social Cost in Congestion Games Igal Milchtaich* December 2016 Congestion models may be studied from either the usersโ point of view or the social one. The first perspective examines the incentives of individual users, who are only interested in their own, personal payoff or cost and ignore the negative externalities that their choice of resources creates for the other users. The second perspective concerns social goals such as the minimization of the mean travel time in a transportation network. This paper studies a more general setting in which individual users attach to the social cost some weight ๐ that is not necessarily 1 or 0, and may also be negative. It examines the comparative-statics question of whether higher ๐ necessarily means higher social welfare at equilibrium. Keywords: Social cost, price of anarchy, congestion games, altruism, stability of equilibrium, potential game JEL Classification code: C72 1 Introduction Negative externalities are a potential source of inefficiency. In congested systems, such a road networks, inefficiency arises when users do not take into account the externalities that their own use of the public resources create for others. For example, slow traffic flow may be the result of drivers only considering the effect of their choice of routes on their own driving time and disregarding its effect on the other drivers. Advising means that would compel users to internalize the social effects of their decisions is a classic theme in economics. Tolls, in particular, may be used for this purpose. However, as a matter of fact, it is not uncommon for people to take social welfare into consideration even in the absence of a material impetus to do so. A simple way to describe such attitudes is to view people as maximizers of a weighted sum of their personal, material payoff and the social payoff. This raises the question of what effect does the weight ๐ attached to the latter has on the actual social welfare at equilibrium. As shown elsewhere (Milchtaich 2012), the effect may actually be negative. In strategic contexts, or games (even symmetric ones), if everyone attaches the same small but positive weight ๐ to each of the other playersโ payoff, the result may paradoxically be lower personal payoffs than in the case ๐ = 0, when only the personal payoffs are considered. The main question this paper asks is whether and under what circumstances such an outcome is possible also in congestion games, where players affect others only through their use of common resources. * Department of Economics, Bar-Ilan University, Ramat Gan 5290002, Israel [email protected] http://faculty.biu.ac.il/~milchti Two lines of research lead to the question outlined above. One line, already mentioned, is the general question of the comparative statics of altruism and spite (Milchtaich 2006a, 2012), that is, the material consequences of attaching an increasingly positive or negative weight ๐, called the altruism coefficient, to the social payoff. The other line of research concerns the effect of altruism or spite on the price of anarchy in congestion games (Chen and Kempe 2008; Hoefer and Skopalik 2009a, 2009b; and others).1 A surprising finding of this research is that the price of anarchy may actually increase with increasing ๐ (Caragiannis et al. 2010; Chen et al. 2014). That is, altruism may lead to system performance deterioration. Superficially, at least, this finding resembles the possibility of โnegativeโ comparative statics mentioned above. However, the two approaches to the study of altruism and spite are fundamentally different. The price of anarchy in a game is the ratio between the largest social cost at any equilibrium and the smallest social cost at any strategy profile. The social cost most commonly considered is the aggregate cost, the total of the playersโ personal costs, which is unaffected by the altruism coefficient ๐. Yet the price of anarchy is affected, because the set of equilibria may chance as ๐ changes. Comparative statics, by contrast, primarily involve local analysis, that is, tracing the effect of a gradual change of the altruism coefficient on the equilibrium behavior at a single, particular equilibrium. Another difference between the two approaches is that the price of anarchy typically concerns not a single game but a family of games, such as all nonatomic congestion games with linear cost functions, for which the price of anarchy is defined as the supremum over all games in the family. For different values of the altruism coefficient ๐, the maximum may be attained at different games. By contrast, comparative statics always concern a single underlying game, with payoffs or costs that depend linearly on the weight ๐ that players attach to the social cost. Therefore, as indicated, they necessarily reflect actual behavioral consequences of this dependence. These differences suggest that the two approaches may give dissimilar, if not outright conflicting, answers to the question of the material consequences of the change in ๐. One goal of this paper is to uncover and analyze the similarities and dissimilarities between them. A central insight emerging from the study of comparative statics of altruism or spite is that a negative relation between the altruism coefficient and the equilibrium level of the social payoff is associated with unstable equilibria. Conversely, stable equilibria guarantee โpositiveโ, or โnormalโ comparative statics (see, in particular, Milchtaich 2012, Theorems 1, 2 and 8). Moreover, this finding holds for any choice of social payoff, not only the aggregate payoff or (the negative of the) cost. It refers to a general notion of static stability, which is applicable to any game and generalizes a number of more special stability concepts (Milchtaich 2016). In general, the stability condition is not implied by the equilibrium condition (and vice versa). However, in some classes of games, the implication does hold, which leads to the question of whether this is so also for certain kinds of congestion games. 1 A related literature examines malicious users in congestion games (e.g., Karakostas and Viglas 2007, Gairing 2009). Malice is similar to spite except that it is restricted to a small number of exceptional users rather than reflecting a โbadโ social norm. 2 In such games, where equilibria are necessarily stable, altruism can only have a positive effect and spite only a negative effect on social welfare. 2 Preliminaries A strategic game โ specifies a (finite or infinite) set of players and, for each player ๐, a strategy set ๐๐ and a payoff function โ๐ : ๐ โ โ, where ๐ = โ๐ ๐๐ is the set of all strategy profiles. A social payoff is any function ๐: ๐ โ โ. An important example in a game with finitely many players is the aggregate payoff, ๐ = โ๐ โ๐ . The altruism coefficient is an exogenously given parameter ๐ โค 1 that specifies a common degree of internalization of the social payoff by all players, reflecting, for example, a similar upbringing or a social norm. It defines a modified game โ๐ , where the payoff of each player ๐ is not the original, personal payoff โ๐ but the modified payoff โ๐๐ = (1 โ ๐)โ๐ + ๐๐. (1) Comparative statics concern the connection between ๐ and the value of ๐ at the Nash equilibria of the modified game โ๐ . In the special case where ๐ is the aggregate payoff, so that โ๐๐ = โ๐ + ๐ โ โ๐ , ๐โ ๐ this connection expresses the effect on the actual equilibrium aggregate personal payoff when each of the players attaches the same weight ๐ to the personal payoff of each of the other players. A positive, negative or zero ๐ may be interpreted as expressing altruism, spite or complete selfishness, respectively. A symmetric ๐-player game is one where all players share the same strategy set ๐ and their payoffs are specified by a single payoff function ๐: ๐ ๐ โ โ that is invariant to permutations of its second through ๐th arguments. If one player uses strategy ๐ฅ and the others use ๐ฆ, ๐ง, โฆ , ๐ค, in any order, the first playerโs payoff is ๐(๐ฅ, ๐ฆ, ๐ง, โฆ , ๐ค). A population game is formally defined as a symmetric two-player game ๐ where the strategy space2 ๐ is a convex set in a (Hausdorff real) linear topological space (for example, the unit simplex in a Euclidean space โ๐ ) and ๐(๐ฅ, ๐ฆ) is continuous in the second argument ๐ฆ for all ๐ฅ โ ๐ (Milchtaich 2012). However, a population game is interpreted not as an interaction between two specific players but as one involving an (effectively) infinite population of individuals who are โplaying the fieldโ. This means that an individualโs payoff ๐(๐ฅ, ๐ฆ) depends only on his own strategy ๐ฅ and on a suitability defined population strategy ๐ฆ โ ๐, which encapsulates the other playersโ actions. A social payoff for a population game is any continuous function ๐: ๐ฬ โ โ whose domain ๐ฬ is the cone of the strategy space ๐. The cone consists of all elements of the form ๐ก๐ฅ, with 2 A strategy space is a strategy set that is endowed with a particular topology, with respect to which continuity and related terms are defined. 3 ๐ฅ โ ๐ and ๐ก > 0. Note that, unlike for ๐-player games, the argument of the social payoff is a single strategy, not a strategy profile. This difference reflects an assumption that, in a population game, the social payoff depends only on the population strategy. The interpretation of a population game ๐ as describing an interaction involving many, individually insignificant players necessitates an adaptation of the notion of internalization of social payoff. The latter is re-interpreted as consideration for the marginal effect of oneโs actions on the social payoff ๐ (Chen and Kempe 2008). To formalize this idea, suppose that ๐ has a directional derivative in every direction ๐ฅฬ โ ๐ฬ and that the derivative depends continuously on the point ๐ฆฬ โ ๐ฬ at which it is evaluated. In other words, the assumption is that the differential ๐๐: ๐ฬ 2 โ โ, defined by (the one-sided derivative) ๐๐(๐ฅฬ, ๐ฆฬ) = ๐ | ๐(๐ก๐ฅฬ + ๐ฆฬ), ๐๐ก ๐ก=0+ exists and is continuous in the second argument (Milchtaich 2012). For altruism coefficient ๐ โค 1, the modified game ๐๐ is then defined by ๐๐ (๐ฅ, ๐ฆ) = (1 โ ๐)๐(๐ฅ, ๐ฆ) + ๐ ๐๐(๐ฅ, ๐ฆ). (2) Comparative statics concern the connection between the altruism coefficient ๐ and the value of the social payoff ๐ at the equilibrium strategies in ๐๐ , that is, those (population) strategies ๐ฆ โ ๐ satisfying ๐๐ (๐ฆ, ๐ฆ) โฅ ๐๐ (๐ฅ, ๐ฆ), ๐ฅ โ ๐. (3) 2.1 Potential games A game โ with a finite number of players is a potential game (Monderer and Shapley 1996) if it admits an (exact) potential (function), that is, a function ๐: ๐ โ โ such that, whenever a single player ๐ changes his strategy, the resulting change in ๐โs payoff equals the change in ๐: โ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โ๐ (๐ฆ) = ๐( ๐ฆ โฃ ๐ฅ๐ ) โ ๐(๐ฆ), ๐ฅ๐ โ ๐๐ , ๐ฆ โ ๐. Here, ๐๐ is player ๐โs strategy set, ๐ is the set of all strategy profiles, and ๐ฆ โฃ ๐ฅ๐ denotes the strategy profile that differs from ๐ฆ only in that player ๐ uses strategy ๐ฅ๐ . The concept of potential game may be adapted to population games, essentially by replacing the difference between the two values of the potential with a derivative (Milchtaich 2012). Specifically, a continuous function ๐ท: ๐ โ โ is an (exact) potential for a population game ๐ with a strategy space ๐ if, for all ๐ฅ, ๐ฆ โ ๐ and 0 < ๐ < 1, the derivative on the left-hand side of the following equality exists and the equality holds: ๐ ๐ท(๐๐ฅ + (1 โ ๐)๐ฆ) = ๐(๐ฅ, ๐๐ฅ + (1 โ ๐)๐ฆ) โ ๐(๐ฆ, ๐๐ฅ + (1 โ ๐)๐ฆ). ๐๐ (4) A useful fact (Milchtaich 2016) is that a sufficient condition for a function ๐ท: ๐ โ โ to be a potential for ๐ is that it can be extended to a continuous function on ๐ฬ, the cone of ๐, such that for the extended function (which is also denoted by ๐ท) the differential ๐๐ท: ๐ฬ 2 โ โ exists, is continuous in the second argument and satisfies 4 ๐๐ท(๐ฅ, ๐ฆ) = ๐(๐ฅ, ๐ฆ), ๐ฅ, ๐ฆ โ ๐. (5) For a game โ with a potential ๐, and any social payoff ๐ and altruism coefficient ๐, the modified game โ๐ is also a potential game, as it is easy to see that the function ๐๐ defined as follows is a potential for it: ๐๐ = (1 โ ๐)๐ + ๐๐. (6) Similarly, for a population game ๐ with a potential ๐ท, and any social payoff ๐ and altruism coefficient ๐, the modified game ๐๐ is also a potential game, as the following function ๐ท ๐ is a potential:3 ๐ท ๐ = (1 โ ๐)๐ท + ๐๐. The next two results are useful for studying the comparative statics of games as above. As their proofs are very similar, only that of the first result is presented. Proposition 1. For a game โ with a potential ๐, a social payoff ๐, and altruism coefficients ๐ and ๐ with ๐ < ๐ โค 1, if two strategy profiles ๐ฆ ๐ and ๐ฆ ๐ are (global) maximum points of ๐๐ and ๐ ๐ , respectively, then ๐(๐ฆ ๐ ) โค ๐(๐ฆ ๐ ). If ๐ฆ ๐ and ๐ฆ ๐ are strict (equivalently, unique) maximum points, then a similar strict inequality holds. Proof. The proof is based on the following identity, which follows immediately from (6): (๐ โ ๐ )(๐(๐ฅ) โ ๐(๐ฆ)) = (1 โ ๐ )(๐๐ (๐ฅ) โ ๐๐ (๐ฆ)) + (1 โ ๐)(๐ ๐ (๐ฆ) โ ๐ ๐ (๐ฅ)). The equality shows that the difference ๐(๐ฅ) โ ๐(๐ฆ) has the opposite sign to that of the expression on the right-hand side. The latter is nonpositive for ๐ฆ = ๐ฆ ๐ and ๐ฅ = ๐ฆ ๐ if these strategy profiles maximize ๐๐ and ๐ ๐ , respectively, and it is negative if they are strict maximum points. โ Proposition 2. For a population game ๐ with a potential ๐ท, a social payoff ๐, and altruism coefficients ๐ and ๐ with ๐ < ๐ โค 1, if two strategies ๐ฆ ๐ and ๐ฆ ๐ are maximum points of ๐ท ๐ and ๐ท ๐ , respectively, then ๐(๐ฆ ๐ ) โค ๐(๐ฆ ๐ ). If ๐ฆ ๐ and ๐ฆ ๐ are strict maximum points, then a similar strict inequality holds. In a game โ with a potential ๐, a global maximum point of ๐ is clearly an equilibrium. In a population game ๐ with a potential ๐ท, even a local maximum point of ๐ท is an equilibrium strategy, and if ๐ท is a convex function, then the converse also holds, in fact, an equilibrium By definition of ๐1 , a condition similar to (5) holds with ๐ท and ๐ replaced by ๐ and ๐1 , respectively. This proves that ๐ (more precisely, its restriction to ๐) is a potential for ๐1 (= ๐๐), which implies that ๐ท๐ is a potential for ๐๐ . 3 5 strategy is necessarily a global maximum point of ๐ท. These assertions follow as immediate conclusions from the following identity, which is a corollary of (4): ๐ | ๐ท(๐๐ฅ + (1 โ ๐)๐ฆ) = ๐(๐ฅ, ๐ฆ) โ ๐(๐ฆ, ๐ฆ). ๐๐ ๐=0+ 2.2 The price of anarchy The price of anarchy is a measure of the distance of the equilibria from the social optimum, which is defined as the maximum of a specified social payoff ๐. Assuming that ๐ is always nonpositive, equivalently, that the social cost โ ๐ is nonnegative, the PoA in a game is the ratio between (i) the value of the social payoff (or social cost) at the equilibrium with the lowest such payoff (or the highest cost) and (ii) the optimal value. In this paper, the focus is on the effect that internalization of the social payoff or cost has on this ratio, in other words, its dependence on the altruism coefficient ๐. This dependence may be studied in a single game or, more generally, for a specified family of games โ, providing that ๐ is meaningful for all games in โ. Thus, the price of anarchy PoA๐ corresponding to altruism coefficient ๐ โค 1 is defined by PoA๐ = sup { ๐(๐ฆ ๐ ) โฃ โ โ โ, ๐ฅ is a strategy profile in โ, โฃ }, ๐(๐ฅ) โฃ ๐ฆ ๐ is an equilibrium in โ๐ (7) where the quotient is interpreted as 1 if the numerator and denominator are both 0 and as โ if only the denominator is 0. A similar definition applies to a family of population games ๐ข, with ๐ replaced by a specified nonpositive social payoff ๐ that is meaningful for all games in ๐ข. 3 Unweighted Congestion Games A (weighted) congestion game has a finite number ๐ of players, numbered from 1 to ๐, who share a finite set ๐ธ of resources (represented, for example, by edges in a graph). Each player ๐ has a weight ๐ค๐ > 0 and a finite collection of strategies. Each strategy of player ๐ is represented by a non-zero vector ๐ฅ๐ = (๐ฅ๐๐ )๐โ๐ธ , where each component ๐ฅ๐๐ is either ๐ค๐ (indicating that using resource ๐ is part of the strategy) or 0 (indicating that the strategy does not include resource ๐). For a strategy profile ๐ฅ, the total use of resource ๐ is expressed โฒ โณ โฒ โณ by the flow, or load, on it ๐ฅ๐ = โ๐ ๐=1 ๐ฅ๐๐ . Two strategy profiles ๐ฅ and ๐ฅ such that ๐ฅ๐ = ๐ฅ๐ for all ๐ are said to be equivalent. The cost of using resource ๐ is given by ๐๐ (๐ฅ๐ ), where ๐๐ : [0, โ) โ โ is a resource-specific cost function. The total cost for each player ๐ is the sum of the costs of the resources included in ๐โs strategy multiplied by the playerโs weight, and the payoff is the negative of the cost. Thus, player ๐โs payoff is given by โ๐ (๐ฅ) = โ โ ๐ฅ๐๐ ๐๐ (๐ฅ๐ ). ๐โ๐ธ The aggregate cost may be viewed as the social cost, and its negative, the aggregate payoff, as the social payoff. Thus, the latter is given by ๐(๐ฅ) = โ โ ๐ฅ๐ ๐๐ (๐ฅ๐ ). ๐โ๐ธ 6 (8) With altruism coefficient ๐ โค 1, the modified payoff of player ๐ is given by โ๐๐ (๐ฅ) = โ โ((1 โ ๐)๐ฅ๐๐ + ๐๐ฅ๐ ) ๐๐ (๐ฅ๐ ). (9) ๐โ๐ธ Note that the expression in parenthesis can be written as ๐ฅ๐๐ + ๐ โ๐โ ๐ ๐ฅ๐๐ . An unweighted congestion game is one where all players ๐ have unity weights, ๐ค๐ = 1. Each strategy is thus a binary vector and the flow on each resource is simply the number of its users, which means that it suffices to specify the values of the cost function only for positive integers. An unweighted congestion game is symmetric if all players have the same strategy set. An unweighted congestion game โ is always a potential game (Rosenthal 1973), with the potential ๐ฅ๐ ๐(๐ฅ) = โ โ โ ๐๐ (๐ก). (10) ๐โ๐ธ ๐ก=1 (If ๐ฅ๐ = 0, the inner sum is 0 by convention.) As in any potential game, the set of equilibria in โ coincides with the set of โlocalโ maximum points of ๐, that is, strategy profiles where no single player can increase ๐ by unilaterally changing his strategy. In particular, it includes the (nonempty) set of all global maximum points of ๐. The following definition and Proposition 3 below identify a condition under which the reverse inclusion also holds, so that the two sets are equal. The definition is inspired by the concept of โflow substitutionโ of Meir Reshef and David C. Parks (personal communication). Definition 1. An unweighted congestion game has the monotonicity property if for every two non-equivalent strategy profiles ๐ฅ โฒ and ๐ฅ โณ there is a strategy profile ๐ฅ that differs from ๐ฅ โฒ only in the strategy of a single player and satisfies min(๐ฅ๐โฒ , ๐ฅ๐โณ ) โค ๐ฅ๐ โค max(๐ฅ๐โฒ , ๐ฅ๐โณ ) for all ๐. The monotonicity property is so called because it is equivalent to the possibility of changing the strategies prescribed by ๐ฅ โฒ sequentially, one player at a time, in such a way that the flow on each resource changes monotonically from its value at ๐ฅ โฒ to the value at ๐ฅ โณ . Lemma 1. The monotonicity property holds if and only if for every two strategy profiles ๐ฅ โฒ and ๐ฅ โณ there are strategy profiles ๐ฅ 0 , ๐ฅ 1 , โฆ , ๐ฅ ๐ such that ๐ฅ 0 = ๐ฅ โฒ , each of the other strategy profiles ๐ฅ ๐ is either identical to its predecessor ๐ฅ ๐โ1 or differs from it only in the strategy of player ๐, ๐ฅ ๐ and ๐ฅ โณ are equivalent, and the sequence of flows ๐ฅ๐0 , ๐ฅ๐1 , โฆ , ๐ฅ๐๐ is monotone for all ๐. Proof. The sufficiency of the condition (โifโ) is proved by setting ๐ฅ = ๐ฅ ๐ , where ๐ is the smallest index for which ๐ฅ ๐ โ ๐ฅ โฒ (which must exist if ๐ฅ โฒ and ๐ฅ โณ are non-equivalent). To prove necessity (โonly ifโ), construct for given strategy profile ๐ฅ โฒ and ๐ฅ โณ a finite sequence ๐ฅฬ 0 , ๐ฅฬ 1 , โฆ as follows: ๐ฅฬ 0 = ๐ฅ โฒ , and for each ๐ โฅ 1, ๐ฅฬ ๐ is a strategy profile that satisfies the conditions specified by Definition 1 for ๐ฅ except that ๐ฅ โฒ there is replaced with ๐ฅฬ ๐โ1 . Note that ๐ฅฬ ๐๐ โ ๐ฅฬ ๐๐โ1 for at least one resource ๐ and that, for all ๐, ๐ฅฬ ๐0 โค ๐ฅฬ ๐1 โค โฏ โค ๐ฅ๐โณ or ๐ฅฬ ๐0 โฅ ๐ฅฬ ๐1 โฅ โฏ โฅ ๐ฅ๐โณ . These facts imply that the sequence cannot be extended indefinitely, because for 7 some ๐พ โฅ 0 the strategy profile ๐ฅฬ ๐พ and ๐ฅ โณ must be equivalent. Now, for 0 โค ๐ โค ๐, define ๐ฅ ๐ as the strategy profile that prescribes the same strategies as ๐ฅฬ ๐พ to all players with index ๐ or lower, and otherwise agrees with ๐ฅ โฒ . It is not difficult to see that, because (๐ฅ๐โฒ =)๐ฅฬ ๐0 , ๐ฅฬ ๐1 , โฆ , ๐ฅฬ ๐๐พ (= ๐ฅ๐โณ ) is monotone for each ๐, the same is true for (๐ฅ๐โฒ =)๐ฅ๐0 , ๐ฅ๐1 , โฆ , ๐ฅ๐๐ (= ๐ฅ๐โณ ). โ Proposition 3. If an unweighted congestion game with nondecreasing cost functions has the monotonicity property, then every equilibrium maximizes the potential ๐. Proof. Consider a game as above that has the monotonicity property. Let ๐ฅ โฒ be an equilibrium, ๐ฅ โณ any other strategy profile, and ๐ฅ 0 , ๐ฅ 1 , โฆ , ๐ฅ ๐ strategy profiles as in Lemma 1. Since ๐ฅ ๐ and ๐ฅ โณ are equivalent, and therefore ๐(๐ฅ ๐ ) = ๐(๐ฅ โณ ), to prove that ๐(๐ฅ โฒ ) โฅ ๐(๐ฅ โณ ) it suffices to show that ๐(๐ฅ ๐ ) โ ๐(๐ฅ ๐โ1 ) is nonpositive for all 1 โค ๐ โค ๐. For each 1 โค ๐ โค ๐, the strategy profile ๐ฅ ๐ can differ from ๐ฅ ๐โ1 only in the strategy of player ๐. Therefore, the difference ๐(๐ฅ ๐ ) โ ๐(๐ฅ ๐โ1 ) is equals to the difference between the playerโs payoff in ๐ฅ ๐ and ๐ฅ ๐โ1 , which is (11) ๐ ๐โ1 โ โ (๐ฅ๐๐ ๐๐ (๐ฅ๐๐ ) โ ๐ฅ๐๐ ๐๐ (๐ฅ๐๐โ1 )). ๐โ๐ธ ๐ ๐โ1 ๐ ๐โ1 In this sum, each term with ๐ฅ๐๐ = ๐ฅ๐๐ is clearly zero. If ๐ฅ๐๐ = 1 and ๐ฅ๐๐ = 0, then it โฒ 0 1 ๐ โฒ ๐โ1 ๐ follows from the monotonicity of (๐ฅ๐ =)๐ฅ๐ , ๐ฅ๐ , โฆ , ๐ฅ๐ that ๐ฅ๐ โค ๐ฅ๐ = ๐ฅ๐ โ 1. Similarly, if ๐ ๐โ1 ๐ฅ๐๐ = 0 and ๐ฅ๐๐ = 1, then ๐ฅ๐โฒ โฅ ๐ฅ๐๐โ1 = ๐ฅ๐๐ + 1. Since ๐๐ is nondecreasing, in all three cases ๐ ๐โ1 ๐ ๐ ๐โ1 ) ๐โ1 โ (๐ฅ๐๐ ๐๐ (๐ฅ๐๐ ) โ ๐ฅ๐๐ ๐๐ (๐ฅ๐๐โ1 )) โค โ (๐ฅ๐๐ ๐๐ (๐ฅ๐โฒ + ๐ฅ๐๐ โ ๐ฅ๐๐ โ ๐ฅ๐๐ ๐๐ (๐ฅ๐โฒ )). The sum over ๐ of the expression on the right is therefore greater than or equal to (11). This sum gives the change in player ๐โs payoff that would result if he deviated directly from ๐ฅ โฒ , by moving from ๐ฅ๐โฒ to ๐ฅ๐๐ . Since ๐ฅ is an equilibrium, this change is nonpositive, with implies the same for (11). โ For an unweighted congestion game โ, consider the social payoff defined as the aggregate payoff ๐ (and the social cost defined as the aggregate cost โ๐). It follows from (6), (8) and (10) that, for altruism coefficient ๐ โค 1, the modified game โ๐ has the potential ๐ฅ๐ ๐ ๐ (๐ฅ) ๐ฅ๐ = โ โ ((1 โ ๐) โ ๐๐ (๐ก) + ๐โ๐ฅ๐ ๐๐ (๐ฅ๐ )) = โ โ โ ๐ฬ๐๐ (๐ก), ๐โ๐ธ ๐ก=1 (12) ๐โ๐ธ ๐ก=1 where the function ๐ฬ๐๐ is defined by ๐ฬ๐๐ = (1 โ ๐)โ๐๐ + ๐โ๐๐ถ๐ and the function ๐๐ถ๐ , the (discrete version of the) marginal social cost, is defined by ๐๐ถ๐ (๐ก) = ๐กโ๐๐ (๐ก) โ (๐ก โ 1)โ๐๐ (๐ก โ 1), ๐ก = 1,2, โฆ. The game โ๐ is generally not a congestion game. However, comparison of (10) with (12) shows that ๐๐ is also the potential of some such game, namely, the unweighted congestion 8 game โฬ๐ that differs from (the original, unmodified one) โ only in that the cost function ๐๐ of each resource ๐ is replaced with ๐ฬ๐๐ . This observation leads to the following comparativestatics result. Theorem 1. Let โ be an unweighted congestion game where the cost functions and marginal social costs are nondecreasing. For the aggregate payoff ๐ as the social payoff, and altruism coefficients ๐ and ๐ with 0 โค ๐ < ๐ โค 1, let ๐ฆ ๐ and ๐ฆ ๐ be equilibria in the modified games โ๐ and โ ๐ , respectively. If โ has the monotonicity property, then ๐(๐ฆ ๐ ) โค ๐(๐ฆ ๐ ). (13) If, in addition, ๐ = 1, then ๐(๐ฆ ๐ ) if the maximum of the social payoff function. Proof. As indicated, the modified games โ๐ and โ ๐ share their potentials, ๐๐ and ๐ ๐ , and therefore also their sets of equilibria, with the unweighted congestion games โฬ๐ and โฬ ๐ . The latter differ from โ only in the cost functions, and therefore have the monotonicity property if and only if โ has it. In this case, it follows from Proposition 3 that the equilibria ๐ฆ ๐ and ๐ฆ ๐ maximize ๐๐ and ๐ ๐ , respectively, because the assumption concerning the costs and marginal social costs in โ implies that ๐ฬ๐๐ and ๐ฬ๐๐ are nondecreasing for every resource ๐. Inequality (13) now follows from Proposition 1. If ๐ = 1, then ๐ ๐ = ๐, so that ๐ฆ ๐ is a maximum point of the social payoff. โ For a game โ with nonnegative costs, Theorem 1 implies that, if the monotonicity property holds, then the price of anarchy (see Section โ2.2) weakly decreases as the players increasingly internalize the social cost of their choice of strategies, and reaches 1 with complete internalization. Combining this observation with a simple bound on the price of anarchy gives the following corollary. Corollary 1. For the family โ of all unweighted congestion games with the monotonicity property that have nonnegative and nondecreasing cost functions and nondecreasing marginal social costs, as well as every subfamily or individual game, the price of anarchy PoA๐ is nonincreasing for 0 โค ๐ โค 1, with 1 โค PoA๐ โค 1/๐. Proof. Consider any ๐ and ๐ with 0 โค ๐ โค ๐ โค 1, game โ โ โ, strategy profile ๐ฅ in โ, and equilibria ๐ฆ ๐ and ๐ฆ ๐ in โ๐ and โ ๐ , respectively. By the nonnegativity and monotonicity of the cost functions, ๐ โค ๐ โค 0, and therefore ๐ โค ๐๐ โค ๐๐. Since ๐ฆ ๐ maximizes ๐๐ (see the proof of Theorem 1), these inequalities give ๐(๐ฅ) โค ๐๐ (๐ฅ) โค ๐๐ (๐ฆ ๐ ) โค ๐๐(๐ฆ ๐ ). Therefore, by (13), ๐(๐ฆ ๐ )/๐(๐ฅ) โค ๐(๐ฆ ๐ )/๐(๐ฅ) โค 1/๐. From theses inequalities the result easily follows. (By definition of PoA๐ , the first inequality gives ๐(๐ฆ ๐ )/๐(๐ฅ) โค PoA๐ .) โ Without the monotonicity property, the conclusion in Corollary 1 does not necessarily hold. Moreover, as the next example shows, this is so even in the special case of a symmetric unweighted congestion game with linear (more precisely, affine) cost functions. Example 1. A symmetric unweighted congestion game โ has two players, four resources and four strategies, which are: resource ๐1 alone, and any two of resources ๐2 , ๐3 and ๐4 . The 9 cost functions are: ๐๐1 (๐ก) = ๐ก + 9, ๐๐2 (๐ก) = ๐ก + 2.5 and ๐๐3 (๐ก) = ๐๐4 (๐ก) = 2๐ก + 1. A strategy profile minimizes the aggregate cost if and only if one player uses ๐2 and ๐3 and the other player uses ๐2 and ๐4 . Such a strategy profile is also an equilibrium in the modified game, for all 0 โค ๐ โค 1. For 0.75 โค ๐ โค 1, there is a second kind of equilibrium, where one player uses ๐1 and the other player uses ๐3 and ๐4 . Therefore, the price of anarchy satisfies PoA๐ = 1 for 0 โค ๐ < 0.75 and is greater than 1 (specifically, = 16/15) for 0.75 โค ๐ โค 1. The game in this example does not have the monotonicity property. For a strategy profile of the first kind (that is, a socially optimal one) the vector of edge flows is (0,2,1,1), and for the second kind, (1,0,1,1). It is not possible to move a single player from his first strategy to another strategy in such a way that the flow on each resource lies between its first and second value, because the flow on the third or fourth resource necessarily changes. Corollary 1 does not hold also when โ is replaced with the whole family of (not necessarily symmetric) unweighted congestion games with cost functions that are linear with nonnegative coefficients. For this family of games (as well as for the subfamily where ๐๐ (๐ก) = ๐ก for all resources ๐), Caragiannis et al. (2010) and Chen et al. (2014) showed that the price of anarchy with respect to the aggregate payoff is strictly increasing for 0 โค ๐ โค 1, and is given by PoA๐ = 5 + 4๐ . 2+๐ Observe that the price of anarchy cannot be strictly increasing for any finite family of games or, in particular, an individual game. With finitely many games, there are only finitely many possible values for the social payoff, which means that PoA๐ can be at most nondecreasing. 3.1 Network congestion games Whether the monotonicity property holds for a symmetric unweighted congestion game only depends on the number of players and the combinatorial structure of the strategy set. In a symmetric unweighted network congestion game (Figure 1), the combinatorial structure reflects the topology of a directed two-terminal network, with origin and destination vertices ๐ and ๐, whose directed edges represent the resources. It is assumed that none of the edges and vertices in the network is โredundantโ, that is, each of them belongs to at least one route from ๐ to ๐. Every such route corresponds to a strategy, which is represented by the binary vector assigning 1 to every edge traversed by the route and 0 to every other edge. An additional assumption concerning symmetric unweighted network congestion games is that all cost functions are nondecreasing. A symmetric unweighted network congestion game can equivalently be described in terms of (i) an undirected twoterminal network ๐บ, where each vertex and each edge belongs to at least one route, and (ii) an assignment of an allowable direction to each edge ๐ in ๐บ, which must be that in which some route in ๐บ traverses ๐ (Milchtaich 2013).4 The set of strategies consists of all allowable routes in ๐บ, that is, routes that traverse all their edges in the allowable directions. It is 4 Not every such assignment results in a directed network as above, as the irredundancy requirement may fail to hold. However, the requirement does hold for at least one assignment. See the Appendix. 10 assumed that at least one such route exists. This alternative description enables classification according to the underlying undirected network: games on the same undirected network ๐บ that differ in the number of players, costs functions or edge directions are grouped together. Such a classification may be useful for identifying certain โtopologicalโ properties of network congestion games, that is, properties of individual games that are necessary consequences of the topology of the undirected network, independently of the other data. Definition 2. An undirected two-terminal network ๐บ has the monotonicity property if all symmetric unweighted network congestion games on ๐บ have the monotonicity property. Lemma 2. If an undirected two-terminal network ๐บ is obtained by connecting two or more networks in series, then ๐บ has the monotonicity property if and only if each of the constituent networks has the property. Proof. Assigning allowable directions to the edges in ๐บ is equivalent to assigning directions in each of the constituent networks ๐บฬ , and a route in ๐บ is allowable if and only if the section that passes through each ๐บฬ is allowable there. Therefore, ๐บ has the monotonicity property if and only if, for every assignment of allowable directions, every number of players and every pair of non-equivalent strategy profiles ๐ฅ โฒ and ๐ฅ โณ , there is a strategy profile ๐ฅ that differs from ๐ฅ โฒ only in the strategy of a single player, who only changes his route in one constituent network ๐บฬ , such that min(๐ฅ๐โฒ , ๐ฅ๐โณ ) โค ๐ฅ๐ โค max(๐ฅ๐โฒ , ๐ฅ๐โณ ) for all ๐ in ๐บฬ . Clearly, this condition holds if and only if every ๐บฬ have the monotonicity property. โ A similar result does not hold for the connection of networks in parallel. Indeed, the monotonicity property does not generally hold even for series-parallel networks, which are those that can be constructed from single edges using only the operations of connecting networks in series or in parallel. Moreover, as the next example shows, in a symmetric unweighted network congestion game on a series-parallel network where the cost functions and marginal social costs are nonnegative and nondecreasing, it is not even generally true that PoA1 = 1 (cf. Corollary 1). By definition of the modified game with ๐ = 1, a strategy profile is an equilibrium in this game if and only if it is โlocallyโ social optimal, that is, any single-player deviation increases the aggregate cost or leaves it unchanged. PoA1 = 1 would mean that this condition automatically implies โglobalโ social optimality, that is, the social cost cannot decrease even if two or more players deviate simultaneously. In an unpublished technical report, Singh (2008) found that this implication holds for all games on a series-parallel network that have cost functions as above. However, the example refutes this finding. Example 2. The series-parallel network in Figure 1a does not have the monotonicity property. For ๐ = 2, a strategy profile ๐ฅ as in Definition 1 does not exist for a strategy profile ๐ฅ โฒ where one player uses the route ๐1 ๐4 and the other uses ๐2 ๐5 and a strategy profile ๐ฅ โณ where the routes are ๐1 ๐5 and ๐3 . Moreover, such ๐ฅ โฒ and ๐ฅ โณ are equilibria in the game specified in Figure 1a, and since in each of them no edge is used by more than one player, they are also โlocalโ minimum points of the aggregate cost. However, the aggregate cost at ๐ฅโฒ is higher than at ๐ฅ โณ , 8 instead of 7. An almost identical pair of equilibria exists in 11 ๐ ๐ ๐1 ๐2 ๐2 ๐1 ๐ ๐1 ๐ ๐2 ๐1 ๐3 ๐3 ๐ฃ ๐4 ๐5 ๐ข ๐ฃ ๐6 ๐3 ๐ข ๐4 ๐5 ๐5 ๐2 ๐ฃ ๐ข ๐4 ๐6 ๐5 ๐4 ๐3 ๐ ๐ ๐ ๐ (a) (b) (c) (d) Figure 1. Symmetric unweighted network congestion games. ๐ต unity-weight players choose routes from ๐ to ๐ . The cost functions are: (a) ๐๐๐ (๐) = ๐๐๐ (๐) = ๐๐ , ๐๐๐ (๐) = ๐๐๐ (๐) = ๐๐ and ๐๐๐ (๐) = ๐๐, (b) the same, and ๐๐๐ (๐) = ๐. ๐๐, (c) ๐๐ (๐) = ๐ for all edges ๐, and (d) any cost functions for which the marginal social costs are nondecreasing. the game in Figure 1b (where the network is not series-parallel). The only difference is that, in ๐ฅ โณ , the first route is ๐1 ๐6 ๐5 . Similarly, in Figure 1c (again, a non-series-parallel network), only the equilibria where the two playersโ routes are ๐1 ๐5 and ๐2 ๐6 are socially optimal. With ๐1 ๐3 ๐6 and ๐2 ๐4 ๐5 , the aggregate cost is higher, but no player can unilaterally decrease it (or his own cost) by choosing a different route. A similar example does not exist for the network in Figure 1d, for any number of players ๐ and any cost functions with nondecreasing marginal social costs. Unlike the other networks in Figure 1, this one is an undirected extension-parallel network (Holzman and Law-Yone 2003).5 This subclass of the series-parallel networks is defined in a similar recursive manner except for the additional restriction that the connection in series of two networks is allowed only if one of them has only one edge (which makes their connection equivalent to the extension of a terminal vertex, ๐ or ๐, in the other network). Proposition 4. An undirected two-terminal network that is extension-parallel, or is obtained by connecting several extension-parallel networks in series, has the monotonicity property. This result is an immediate corollary of Lemma 2 and the following one, which shows that extension-parallel networks actually satisfy a stronger condition than the monotonicity property. A network ๐บ has the augmented monotonicity property (AMP) if the augmented network ๐บ โ obtained by connecting ๐บ in parallel with a single edge ๐ โ has the monotonicity property. The edge ๐ โ may be interpreted as representing a common outside option, an alternative to using the network ๐บ. 5 The original meaning of extension-parallel networks concerned directed networks. An alternative term for the undirected version of these networks, which is the version considered here, is networks with linearly independent routes. These undirected two-terminal networks are characterized by the property that each route includes at least one edge that does not belong to any other route (Milchtaich 2006b). 12 Lemma 3. An undirected two-terminal network has the augmented monotonicity property if and only if it is extension-parallel. Proof. One direction of the proof is by induction on the structure of extension-parallel networks. For a one-edge network, the AMP holds trivially. If a network ๐บ is obtained by connecting in series an extension-parallel network ๐บฬ that has the AMP and a single edge ๐, then ๐บ also has the AMP. This is because the set of routes (equivalently, allowable routes; see Corollary A2 in the Appendix, condition (ii)) in ๐บ is obtained from that in ๐บฬ simply by the addition of ๐ (and its non-terminal end vertex) to each route. Therefore, the flow on ๐ and the flow on ๐ โ sum up to the number of players ๐ for every strategy profile in the augmented network ๐บฬ โ , so that the required pair of inequalities in Definition 1 holds automatically for the first edge if it holds for the second edge. It remains to consider a network ๐บ obtained by connecting in parallel two extension-parallel networks ๐บฬ and ๐บฬ that have the AMP, and show that the augmented network ๐บ โ (consisting of ๐บฬ , ๐บฬ and a single edge ๐ โ all connected in parallel) has the monotonicity property. For any number of players ๐, consider two non-equivalent strategy profiles ๐ฅโฒ and ๐ฅ โณ where the players use routes in ๐บ โ . Let ๐ฅฬ โฒ and ๐ฅฬ โณ be the strategy profiles obtained from ๐ฅโฒ and ๐ฅ โณ by replacing each route that lies outside ๐บฬ โ (that is, in ๐บฬ ) with the single-edge route ๐ โ . In particular, for ๐ฅโฒ, ๐ฅฬ ๐โฒ โ gives the number of players who are not using a route in ๐บฬ , and similarly for ๐ฅ โณ and ๐ฅฬ ๐โณโ . Since ๐บฬ has the AMP, if ๐ฅฬ โฒ and ๐ฅฬ โณ are non-equivalent, then applying Definition 1 to them (instead of ๐ฅโฒ and ๐ฅ โณ ) gives a strategy profile ๐ฅฬ , which only involves routes in ๐บฬ โ . For ๐บฬ , the strategy profiles ๐ฅฬ โฒ , ๐ฅฬ โณ and (if the last two are non-equivalent) ๐ฅฬ are constructed analogously. Using these ingredients, a strategy profile ๐ฅ satisfying the conditions in Definition 1 is constructed below. Note, first, that the strategy profiles ๐ฅฬ โฒ and ๐ฅฬ โณ are non-equivalent or ๐ฅฬ โฒ and ๐ฅฬ โณ are nonequivalent. Otherwise, both the flow on each edge in ๐บฬ and the total number of players using routes in that network (which is ๐ minus the number of players not doing so) would be the same for ๐ฅโฒ and ๐ฅ โณ , and similarly for ๐บฬ . However, the conclusion implies that the number of players using ๐ โ is also the same for ๐ฅโฒ and ๐ฅ โณ , which contradicts their assumed non-equivalence. By symmetry, it suffices to consider the case where ๐ฅฬ โฒ and ๐ฅฬ โณ are nonequivalent, so that a strategy profile ๐ฅฬ as above and a single player ๐ with ๐ฅฬ ๐ โ ๐ฅฬ ๐โฒ exist. Suppose, first, that ๐ฅฬ ๐โฒ is a route in ๐บฬ (hence, ๐ฅฬ ๐โฒ = ๐ฅ๐โฒ ). If ๐ฅฬ ๐ is also a route there, then define ๐ฅ by ๐ฅ๐ = ๐ฅฬ ๐ . (The other strategies in ๐ฅ are as in ๐ฅ โฒ .) If ๐ฅฬ ๐ is ๐ โ , then ๐ฅฬ ๐โณโ > ๐ฅฬ ๐โฒ โ by definition of the monotonicity property. If in addition ๐ฅฬ๐โณโ โฅ ๐ฅฬ๐โฒ โ , then necessarily ๐ฅ๐โณโ > ๐ฅ๐โฒ โ , and in this case, set ๐ฅ๐ to be ๐ โ . If ๐ฅฬ๐โณโ < ๐ฅฬ๐โฒ โ , then ๐ฅฬ โฒ and ๐ฅฬ โณ are not equivalent and there is a player ๐ with ๐ฅฬ๐ โ ๐ฅฬ๐โฒ where ๐ฅฬ๐ is not ๐ โ (but rather a route in ๐บฬ ). In this case, define ๐ฅ by ๐ฅ๐ = ๐ฅฬ๐ or ๐ฅ๐ = ๐ฅฬ๐ if ๐ฅฬ๐โฒ is or is not ๐ โ , respectively. Thus, in the first case, player ๐ moves to the route that player ๐ (who may or may not be ๐) was supposed to move to. In remains to consider the case where ๐ฅฬ ๐โฒ is ๐ โ and ๐ฅฬ ๐ is a route in ๐บฬ , which by definition of the monotonicity property implies that ๐ฅฬ ๐โณโ < ๐ฅฬ ๐โฒ โ . If in addition ๐ฅฬ๐โณโ โค ๐ฅฬ๐โฒ โ , then necessarily ๐ฅ๐โณโ < ๐ฅ๐โฒ โ , and in this case, choose any player ๐ for whom ๐ฅ๐โฒ is ๐ โ and define ๐ฅ by ๐ฅ๐ = ๐ฅฬ ๐ . If ๐ฅฬ๐โณโ > ๐ฅฬ๐โฒ โ , then ๐ฅฬ โฒ and ๐ฅฬ โณ are not equivalent and there is a player ๐ with ๐ฅฬ๐ โ ๐ฅฬ๐โฒ where ๐ฅฬ๐โฒ is 13 a route in ๐บฬ . In this case, set ๐ฅ๐ = ๐ฅฬ ๐ or ๐ฅ๐ = ๐ฅฬ๐ if ๐ฅฬ๐โฒ is or is not ๐ โ , respectively. This completes the first part of the proof: all undirected extension-parallel networks have the augmented monotonicity property. The reason why these networks are the only ones having the AMP is that one (or more) of three specific undirected networks is embedded in every undirected network ๐บ that is not extension-parallel (in other words, a network with linearly dependent routes; see footnote 5). These three networks are: the figure-eight network, which is the network in Figure 1a minus the edge ๐3 , the โexpandedโ figure-eight network obtained from the previous network by replacing the vertex ๐ฃ with two vertices connected by an edge, and the Wheatstone network, which is the undirected version of the network in Figure 1b minus ๐3 (Milchtaich 2006b, Proposition 4). Therefore, connecting ๐บ in parallel with a single edge ๐ โ results in an undirected network ๐บ โ for which a two-player game as in Example 2, that is, a game without the monotonicity property, exists. โ The (weaker) monotonicity property itself is not limited to undirected extension-parallel networks and those, such as the figure-eight network, that are obtained by connecting several extension-parallel networks in series. It also holds for certain networks that are not even series-parallel, in particular, the Wheatstone network. This can be seen by comparing (i) the directed Wheatstone network obtained by deleting edge ๐3 in Figure 1c with (ii) the network resulting from deleting edge ๐3 in Figure 1d. Network (ii) is extension-parallel, and therefore has the monotonicity property. The addition of ๐6 (and ๐ฃ) to each route in this network that includes either ๐2 or ๐4 defines a one-to-one correspondence between the set of routes in (ii) and the set of allowable routes in (i). As the flow on ๐6 must always be equal to the number of players ๐ minus the flow on ๐5 , it is easy to conclude that network (i) also has the monotonicity property. The last comment, Proposition 3 and Lemmas 2 and 3 together prove the following result, which is a slightly stronger version of one originally proved by Fotakis (2010, Lemma 4, which the author notes is also implicit in the work of Holzman and Law-Yone 1997). Proposition 5. In a symmetric unweighted network congestion game on an undirected network with the monotonicity property, in particular, the Wheatstone network or an extension-parallel one or a network obtained by connecting several such networks in series, every equilibrium is a (global) maximum point of the potential. Undirected extension-parallel networks are special also with respect to two alternative notions of social optimality, or efficiency. One interesting property of an undirected network ๐บ is that, in every symmetric unweighted network congestion game on ๐บ, every equilibrium is a strong equilibrium. Holzman and Law-Yone (1997, 2003) show that an undirected two-terminal network has this property if and only if it is extension-parallel.6 An equivalent way of stating this result is that an extension-parallel network is a necessary and 6 The authors actually establish their results for directed networks (and nonnegative costs). However, it is not very difficult to conclude from these results that the version given here also holds, for example, by using Proposition A1 in the Appendix. 14 sufficient condition for weak Pareto efficiency of all equilibria in all corresponding games, so that it is never possible to make everyone better off by altering the playersโ equilibrium route choices. The equivalence holds because an equilibrium is strong if and only if the strategy choices of every group of players constitute a weak Pareto efficient equilibrium in the subgame defined by fixing the strategies of the remaining players. That subgame is itself a symmetric unweighted network congestion game on the same network. Epstein et al. (2009) call an undirected two-terminal network ๐บ efficient if, in every symmetric unweighted network congestion game on ๐บ, every equilibrium minimizes the makespan, that is, it is optimal with respect to the maximum (rather than the sum) of the playersโ costs. They show that an undirected network is efficient if and only if it is extension-parallel. Kuniavsky and Smorodinsky (2013) study the relation between the set of equilibria in a symmetric unweighted congestion game and its set of greedy strategy profiles. The latter consists of all strategy profiles that can be obtained in a dynamic setting where players join the game sequentially according to some pre-determined order and each player, upon arrival, myopically chooses a strategy that best responds to his predecessorsโ choices. Their main result (Theorem 4.1) implies that the above two sets coincide in all symmetric unweighted network congestion games on an undirected two-terminal network ๐บ if and only if ๐บ is extension-parallel. 4 Congestion Games with Splittable flow A congestion game with splittable flow is obtained from a (weighted) congestion game as in Section โ3 (hereafter, a game with unsplittable flow) by replacing the strategy set of each player ๐ with its convex hull. Thus, each strategy ๐ฅ๐ = (๐ฅ๐๐ )๐โ๐ธ is a convex combination of strategies as in Section โ3 (hereafter, โpureโ strategies), with coefficients that express the fraction of the playerโs weight ๐ค๐ โshippedโ on each of them. The resulting amount of use by player ๐ of each resource ๐ is given by the corresponding component 0 โค ๐ฅ๐๐ โค ๐ค๐ . An interesting special case is that of linear congestion games with splittable flow, where there are constants ๐๐ > 0 and ๐๐ for each resource ๐ such that ๐๐ (๐ก) = ๐๐ ๐ก + ๐๐ . The linearity of the cost functions implies that the game is a potential game, as it is not difficult to check that the function ๐ defined as follows (Fotakis et al. 2005) is an (exact) potential: ๐(๐ฅ) = โ โ (๐๐ ๐โ๐ธ 2 ๐ฅ๐2 + โ๐ ๐=1 ๐ฅ๐๐ + ๐๐ ๐ฅ๐ ). 2 Observe that ๐ โฅ ๐, where ๐ is the aggregate payoff (8). With the latter as the social payoff, and altruism coefficient ๐ โค 1, the modified game has the following potential (see (6)): ๐๐ (๐ฅ) = โ โ (๐๐ ๐โ๐ธ 2 (1 + ๐)๐ฅ๐2 + (1 โ ๐) โ๐ ๐=1 ๐ฅ๐๐ + ๐๐ ๐ฅ๐ ). 2 Clearly, there exists at least one strategy profile maximizing ๐๐ , and every such strategy 15 (14) profile is an equilibrium in the modified game. The next theorem, which shows that the equilibrium is in fact unique if ๐ โ 1, examines the dependence of the corresponding social payoff on the altruism coefficient. Theorem 2. For a linear congestion game with splittable flow โ, and the aggregate payoff ๐ as the social payoff, the modified game โ๐ has a unique equilibrium ๐ฆ ๐ for every โ1 โค ๐ < 1, and the function ๐ โฆ ๐ฆ ๐ is continuous. For โ1 , uniqueness does not necessarily hold but every equilibrium ๐ฆ1 maximizes ๐. The function ๐: [โ1,1] โ โ (well) defined by ๐(๐) = ๐(๐ฆ ๐ ) is absolutely continuous and there is a partition of [โ1,1] into finitely many intervals such that ๐ is either constant or strictly increasing in each interval. Proof. To prove that for โ1 โค ๐ < 1 there is only one equilibrium in โ๐ , and therefore only one maximum point of the potential ๐๐ , it suffices to show that every equilibrium ๐ฆ is a strict global maximum point of ๐๐ . Thus, it has to be shown that for every strategy profile ๐ฅ โ ๐ฆ the function ฮ: [0,1] โ โ defined by ฮ(ฮป) = ๐๐ (ฮป๐ฅ + (1 โ ๐)๐ฆ) satisfies ฮ(0) > ฮ(ฮป) for all 0 < ฮป โค 1. For this, it suffices that ฮโฒ (0) โค 0 and ฮโณ < 0. Substitution in (14) and derivation give: ๐ โฒ (0) ฮ = โ โ โ(๐๐ (1 + ๐)๐ฆ๐ + ๐๐ (1 โ ๐)๐ฆ๐๐ + ๐๐ )(๐ฅ๐๐ โ ๐ฆ๐๐ ). (15) ๐=1 ๐โ๐ธ The expression on the right-hand side is equal to the sum ๐ โ ๐=1 ๐ | โ๐ ( ๐ฆ โฃ ฮป๐ฅ๐ + (1 โ ๐)๐ฆ๐ ), ๐ฮป ฮป=0+ ๐ whose terms are all nonpositive because ๐ฆ is an equilibrium in โ๐ . This proves that ฮโฒ (0) โค 0. The second inequality above holds even without the equilibrium assumption (which shows that the potential ๐๐ is a strictly concave function on the set of strategy profiles): ๐ โณ ฮ = โ(1 โ ๐) โ ๐๐ โ(๐ฅ๐๐ โ ๐ฆ๐๐ )2 โ (1 + ๐) โ ๐๐ (๐ฅ๐ โ ๐ฆ๐ )2 < 0. ๐โ๐ธ ๐=1 ๐โ๐ธ For ๐ = 1, ฮโณ โค 0 (and equality holds if and only if the flows on all resources are the same for ๐ฅ and ๐ฆ). Therefore, every equilibrium in โ1 is a maximum point of the potential ๐1 (which coincides with ๐). It follows that the projection on the first two coordinates of the set ๐ = { (๐, ๐ผ, ๐ฆ) โฃ โ1 โค ๐ โค 1, ๐ผ = ๐(๐ฆ), ๐ฆ is an equilibrium in โ๐ } is the graph of a function, denoted ๐: [โ1,1] โ โ. By the TarskiโSeidenberg theorem, a sufficient condition for ๐ to be continuous and semialgebraic is that the set ๐ is closed (equivalently, compact) and semialgebraic. This condition holds because, as shown, ๐ฆ is an equilibrium in โ๐ if and only if it maximizes ๐๐ , hence if and only if the (polynomial) expression in (15) is nonpositive for all profiles ๐ฅ of pure strategies (which, by linearity, 16 ๐1 ๐3 ๐2 ๐3 ๐1 ๐2 Figure 2. A three-player congestion game with splittable flow. Each player ๐ has to ship a unit flow from ๐๐ to ๐ ๐ , possibly through multiple routes. The cost per unit flow for each edge is equal to the total flow of it, except for the three edges marked by think lines, where there is an additional (per unit) fixed cost of ๐. automatically implies the same for mixed strategies). The compactness of ๐ also implies that, on the half-open interval [โ1,1), the function mapping each ๐ to the corresponding unique equilibrium ๐ฆ ๐ is continuous. Since ๐ is a continuous semialgebraic function, for every โ1 โค ๐ โค 1 there is some ๐ > 0, a pair of positive integers ๐๐ and ๐๐ฟ and a pair of analytic functions ๐๐ , ๐๐ฟ : (โ๐, ๐) โ โ such that ๐(๐ ) = ๐๐ ((๐ โ ๐)1/๐๐ ) for all ๐ โค ๐ โค min{1, ๐ + ๐ ๐๐ /2} and ๐(๐ ) = ๐๐ฟ ((๐ โ ๐ )1/๐๐ฟ ) for all max{โ1, ๐ โ ๐ ๐๐ฟ /2} โค ๐ โค ๐ (see Bochnak et al. 1998, Proposition 8.1.12). By decreasing ๐, if necessarily, it can be guaranteed that each of the analytic functions ๐๐ and ๐๐ฟ is either constant or strictly monotone. The collection of all intervals of the form (๐ โ ๐ ๐๐ฟ /2, ๐ + ๐ ๐๐ /2), where ๐ varies over all points in [โ1,1] and ๐, ๐๐ and ๐๐ฟ vary correspondingly, constitutes an open cover of [โ1,1]. Any finite subcover yields a finite set of points โ1 = ๐0 < ๐1 < โฏ < ๐๐ฟ = 1 such that, for all 1 โค ๐ โค ๐ฟ, the restriction of ๐ to the interval [๐๐โ1 , ๐๐ ] is continuous and either constant or strictly monotone and has a continuous derivative in (๐๐โ1 , ๐๐ ). These properties imply that ๐ is absolutely continuous in the interval. It remains to observe that, by Proposition 1, it cannot be strictly decreasing there. โ The next example illustrates the theorem. Example 3. In a three-player linear congestion game with splittable flow, ๐ธ is the set of all edges in the undirected graph shown in Figure 2. The weight of each player ๐ is 1, and the playerโs โpureโ strategies are the (simple) paths in the graph that connect ๐โs origin vertex ๐๐ with the destination vertex ๐๐ .7 The cost of each edge ๐ is equal to the flow on it ๐ฅ๐ , except 7 Note that this (asymmetric) game differs from the network congestion game considered in Section โ3.1 in that edges can be traversed in both directions. 17 for the three โthickโ edges, for which the cost is ๐ฅ๐ + 2. The minimal total cost of 9 is achieved when all players use the short, three-edge path connecting their origin and destination vertices. If the altruism coefficient satisfies 1/3 โค ๐ โค 1, then this strategy profile is also the unique equilibrium in the modified game, where the payoffs are given by (9). However, for a lower altruism coefficient, โ1 โค ๐ < 1/3, it is not an equilibrium. Instead, the unique equilibrium is for every player to ship only (9 โ ๐)/(10 โ 4๐) of the weight on the short path, and the rest on the longer, four-edge path. The corresponding social payoff ๐(๐), which is the negative of the aggregate cost, is given by ๐(๐) = โ (169 ( 1โ๐ 2 ) + 8). 10 โ 4๐ As ๐ increases from โ1 to 1/3, ๐(๐) increases to the maximum social payoff. Put differently, as the players increasingly internalize the social cost, the latter decreases, until it reaches its minimum and no longer changes. Thus, the price of anarchy is nonincreasing. It follows immediately from Theorem 2 that a nonincreasing price of anarchy with respect to the aggregate payoff is a feature not only of individual games as in Example 3, where the cost functions are nonnegative (equivalently, ๐๐ โฅ 0 for every resource ๐), but also of the whole family of such games. The following corollary combines this observation with an extension of the result that, without altruism (that is, for ๐ = 0), the price of anarchy for games as above does not exceed 3/2 (Cominetti et al. 2009). Corollary 2. For the family โ of all linear congestion games with splittable flow and nonnegative cost functions, as well as every subfamily or individual game, the price of anarchy PoA๐ is nonincreasing for โ1 โค ๐ โค 1, with 1 โค PoA๐ โค 1 1 + . 1+๐ 2 (16) Proof. Consider, more generally, any ๐-player congestion game with splittable flow โ where every cost function ๐๐ is differentiable, nonnegative and nondecreasing. It follows as an immediate conclusion from Theorem 3.1 and Proposition 3.2 in Cominetti et al. (2009) that the aggregate cost โ๐ satisfies (1 โ max ๐ฃ๐ (๐๐ ))(โ๐(๐ฆ)) โค โ๐(๐ฅ) ๐ (17) for any equilibrium ๐ฆ, any strategy profile ๐ฅ and any assignment of a number ๐ฃ๐ (๐๐ ) to each cost function ๐๐ such that ๐ โ(๐๐ (๐ก) โ ๐๐ (๐ )) + (๐ 2 /4 โ (๐ก โ ๐ /2)2 /๐)โ๐๐โฒ (๐ก) . ๐ก๐๐ (๐ก) ๐ ,๐ก>0 ๐ฃ๐ (๐๐ ) โฅ sup (18) A straightforward generalization of the proofs of these results yields a similar conclusion in the more general case where ๐ฆ is an equilibrium in the modified game โ๐ , for any โ1 โค ๐ โค 1. The only difference is that the numerator in inequality (18) is replaced with ๐ โ(๐๐ (๐ก) โ ๐๐ (๐ )) + (๐(๐ โ ๐ก)๐ก + (1 โ ๐)(๐ 2 /4 โ (๐ก โ ๐ /2)2 /๐))โ๐๐โฒ (๐ก). 18 In the linear case, where the derivative ๐๐โฒ is constant8 and therefore ๐๐ (๐ก) โฅ ๐ก๐๐โฒ , the modified inequality holds for (๐ โ(๐ก โ ๐ ) + ๐(๐ โ ๐ก)๐ก + (1 โ ๐)(๐ 2 /4 โ (๐ก โ ๐ /2)2 /๐))โ๐๐โฒ ๐ก 2 ๐๐โฒ ๐ ,๐ก>0 1โ๐ = sup(๐พโ(1 โ ๐พ) + ๐(๐พ โ 1) + (1 โ ๐)(๐พ 2 /4 โ (1 โ ๐พ/2)2 /๐)) = . 3 + ๐ + 4/(๐ โ 1) ๐พ>0 ๐ฃ๐ (๐๐ ) = sup Substitution in (17) and rearrangement give 1 1 โ๐(๐ฆ) โค ( + ) (โ๐(๐ฅ)). 1 + ๐ + (1 โ ๐)/๐ 2 โ This inequality proves (16). Corollary 2 points to a striking difference between unweighted linear congestion games with splittable and unsplittable flow. For the latter, as indicated in Section โ3, PoA๐ is strictly increasing for 0 โค ๐ โค 1. 5 Nonatomic Congestion Games A nonatomic congestion game has an infinite set, or population, of players ๐ผ (e.g., the unit interval [0,1]), endowed with a nonatomic probability measure ๐, the population measure (e.g., Lebesgue measure). The players share a finite set ๐ธ of resources and a (finite) collection of (โpureโ) strategies, which are non-zero binary vectors of the form ๐ = (๐๐ )๐โ๐ธ . A strategy profile is any assignment of a strategy ๐(๐) = (๐๐ (๐))๐โ๐ธ to each player ๐ such that, for each resource ๐, the set of all players ๐ with ๐๐ (๐) = 1 is measurable. The flow on resource ๐, ๐ฆ๐ , is the measure of this set, and is therefore given by ๐ฆ๐ = โซ ๐๐ (๐) ๐๐(๐). The vector ๐ฆ = (๐ฆ๐ )๐โ๐ธ is the population strategy. It lies in the convex hull of the set of strategies, denoted ๐, which may be thought of as the collection of all mixed strategies. The cost of using each resource ๐ is given by ๐๐ (๐ฆ๐ ), where ๐๐ : [0, โ) โ โ is a strictly increasing and continuously differentiable cost function. The cost of using a strategy ๐ = (๐๐ )๐โ๐ธ is the sum of the costs of all resources ๐ with ๐๐ = 1, and the associated payoff is the negative of the cost. A natural extension to mixed strategies assigns to each ๐ฅ = (๐ฅ๐ )๐โ๐ธ โ ๐ the payoff ๐(๐ฅ, ๐ฆ) = โ โ ๐ฅ๐ ๐๐ (๐ฆ๐ ). (19) ๐โ๐ธ The mean payoff ๐(๐ฆ) is defined by ๐(๐ฆ) = ๐(๐ฆ, ๐ฆ) = โ โ ๐ฆ๐ ๐๐ (๐ฆ๐ ). (20) ๐โ๐ธ 8 For linear congestion games with splittable flow as defined in this paper, the derivative is a positive constant, ๐๐ > 0. However, the bound (16) and its proof here apply also if ๐๐ = 0 is allowed for. 19 Eq. (19) defines a population game ๐ (see Section โ2), where the strategy space is ๐. The mean payoff ๐ is a corresponding social payoff function, as the expression on the right-hand side of (20) (as well as that in (19)) is meaningful also for elements of ๐ฬ, the cone of ๐. With altruism coefficient 0 โค ๐ โค 1, the modified game ๐๐ (which is defined by (2)) is given by ๐๐ (๐ฅ, ๐ฆ) = โ โ ๐ฅ๐ ๐ฬ๐๐ (๐ฆ๐ ) = โ โ ๐ฅ๐ (๐๐ (๐ฆ๐ ) + ๐โ๐ฆ๐ ๐๐โฒ (๐ฆ๐ )), ๐โ๐ธ (21) ๐โ๐ธ where ๐ฬ๐๐ = (1 โ ๐)โ๐๐ + ๐โ๐๐ถ๐ and ๐๐ถ๐ : [0, โ) โ โ is the marginal social cost defined by ๐๐ถ๐ (๐ก) = ๐ ๐ก๐ (๐ก) = ๐๐ (๐ก) + ๐กโ๐๐โฒ (๐ก). ๐๐ก ๐ (22) If ๐๐ถ๐ is strictly increasing and continuously differentiable for every resource ๐, then ๐๐ may also be viewed as the population game representing a nonatomic congestion game, which differs from the original one in that each cost function ๐๐ is replaced with ๐ฬ๐๐ . A sufficient condition for ๐๐ถ๐ to be strictly increasing in any interval is that ๐๐ is convex there. The population game ๐ defined above is a potential game, as it is easy to see that the (well known; see Milchtaich 2004, Section 5) function ๐ท: ๐ฬ โ โ defined by ๐ฅฬ๐ ๐ท(๐ฅฬ) = โ โ โซ ๐๐ (๐ก) ๐๐ก ๐โ๐ธ 0 satisfies condition (5) and is therefore a potential. Since the cost functions ๐๐ are strictly increasing, ๐ท is strictly concave, and its restriction to the strategy space ๐ therefore has a unique maximum point. If the marginal social costs are strictly increasing in [0,1], then the mean payoff ๐ is also strictly concave. With 0 โค ๐ โค 1, the same then holds for ๐ท ๐ = (1 โ ๐)๐ท + ๐๐, which as indicated (see Section โ2.1) is a potential for ๐๐ . By the remarks that follow Proposition 2, the unique maximum point of ๐ท ๐ is the unique equilibrium strategy in ๐๐ . This conclusion and Proposition 2 itself give the following result. Proposition 6. Consider a nonatomic congestion game where for each resource ๐ the marginal social cost ๐๐ถ๐ is strictly increasing in [0,1]. For the corresponding population game ๐, with strategy space ๐, and the mean payoff ๐ as the social payoff, the modified game ๐๐ has a unique equilibrium strategy ๐ฆ ๐ for every 0 โค ๐ โค 1. For 0 โค ๐ < ๐ โค 1, ๐(๐ฆ ๐ ) โค ๐(๐ฆ ๐ ) โค ๐(๐ฆ1 ) = max ๐(๐ฅ). ๐ฅโ๐ Proposition 6 shows that nonatomic congestion games, like the linear congestion games with splittable flow studied in Section โ4, always exhibit โnormalโ comparative statics. The next result provides a more specific description in a special case. Theorem 3. Consider a nonatomic congestion game where for each resource ๐ the cost function ๐๐ is a polynomial and ๐๐ถ๐ is strictly increasing in [0,1]. For the corresponding 20 population game ๐, and the mean payoff ๐ as the social payoff, the function that assigns to each 0 โค ๐ โค 1 the unique equilibrium strategy ๐ฆ ๐ in the modified game ๐๐ is continuous. The function ๐: [0,1] โ โ defined by ๐(๐) = ๐(๐ฆ ๐ ) is absolutely continuous and there is a partition of [0,1] into finitely many intervals such that ๐ is either constant or strictly increasing in each interval. Proof. In view of Proposition 6, the graph of the function ๐ is the projection on the first two coordinates of the set ๐ = { (๐, ๐ผ, ๐ฆ) โฃ 0 โค ๐ โค 1, ๐ผ = ๐(๐ฆ), ๐ฆ is an equilibrium strategy in ๐๐ }. The equilibrium condition for ๐ฆ is expressed by (3). Since ๐๐ (๐ฅ, ๐ฆ) is linear in the first argument ๐ฅ (see (21)), the condition holds if and only if the inequality in (3) holds for every ๐ฅ that is a pure strategy. As there are only finitely many pure strategies, ๐ is a semialgebraic compact set. It follows that ๐ is continuous and semialgebraic and that the function ๐ โฆ ๐ฆ ๐ is continuous. The other assertions concerning ๐ now follow by the same arguments used in the last part of the proof of Theorem 2, except that they rely on Proposition 6 rather than 1. โ Another special case of Proposition 6 is the following corollary, whose proof is similar to that of Corollary 1. Corollary 3. Consider the family of all nonatomic congestion games where for each resource ๐ the cost function ๐๐ is a polynomial with nonnegative coefficients. For the corresponding family of population games ๐ข, as well as every subfamily or individual game, the price of anarchy PoA๐ is nonincreasing for 0 โค ๐ โค 1, with 1 โค PoA๐ โค 1/๐. The bound PoA๐ โค 1/๐, which was first established by Chen and Kempe (2008), can be improved upon. These authors (see also Reshef and Parks 2015) show that, for the subfamily where the cost functions are polynomials of degree at most ๐ โฅ 1, the price of anarchy can be determined precisely. For 0 โค ๐ โค 1, 1 โ1 1 + ๐๐ 1+๐ ๐ ) ) . PoA = (1 + ๐๐ โ ๐ ( 1+๐ As ๐ increases, this expression monotonically increases and tends to the limit (hence, the least upper bound) (๐(1 โ ln ๐)) al. 2014 mistakenly assert). โ1 (and not to 1/๐, as Chen and Kempe 2008 and Chen et 6 Comparative Statics and Stability Comparative statics are closely linked with the stability or instability of the concerned equilibria or equilibrium strategies in the corresponding modified games (Milchtaich 2012). More specifically, stability is associated with normal, โpositiveโ comparative statics, whereby an increase in the altruism coefficient increases social welfare, and (definite) instability is 21 associated with โnegativeโ comparative statics, in which the opposite relation holds. This association, which is detailed below, provides a general context to the results obtained in the previous sections, as it does not depend on the existence of a potential. The notion of stability relevant for comparative statics is static stability (Milchtaich 2016). This concept differs from dynamic stability in not involving any extraneous law of motion. Instead, stability is completely determined by the game itself, that is, by the playersโ payoffs. In symmetric games, static stability generalizes a number of more special stability concepts such as evolutionarily stable strategy, or ESS. It is a local concept, defined with respect to the topology specified for the playersโ common strategy space ๐.9 In principle, any topology may be used, but in many games, there is a single natural one. Definition 3. A strategy ๐ฆ in a symmetric ๐-player game with payoff function ๐ is stable, weakly stable or definitely unstable if it has a neighborhood where, for every strategy ๐ฅ โ ๐ฆ, the inequality ๐ 1 โ(๐(๐ฅ, ๐ฅ, โโฆ , ๐ฅ , ๐ฆ, โโฆ , ๐ฅ , ๐ฆ, โโฆ , ๐ฆ ) โ ๐(๐ฆ, ๐ฅ, โโฆ , ๐ฆ )) < 0, ๐ ๐โ1 times ๐โ1 times ๐=1 ๐โ๐ times (23) ๐โ๐ times a similar weak inequality or the reverse (strict) inequality, respectively, holds. If the corresponding inequality holds for all strategies ๐ฅ โ ๐ฆ,10 then ๐ฆ is said to be globally stable, weakly stable or definitely unstable, respectively. One interpretation of (23) is that moving all players one-by-one from ๐ฆ to the nearby strategy ๐ฅ harms them on average. For population games, stability of a strategy is defined differently than for symmetric games, reflecting the different interpretation of these games as describing a large crowd of interacting individuals. Essentially, the sum in (23) is replaced with an integral. Definition 4. A strategy ๐ฆ in a population game ๐ is stable, weakly stable or definitely unstable if it has a neighborhood where, for every strategy ๐ฅ โ ๐ฆ, the inequality 1 โซ (๐(๐ฅ, ๐๐ฅ + (1 โ ๐)๐ฆ) โ ๐(๐ฆ, ๐๐ฅ + (1 โ ๐)๐ฆ)) ๐๐ < 0, 0 a similar weak inequality or the reverse (strict) inequality, respectively, holds. Formally, at least, Definition 3 also induces a notion of stability of strategy profiles in general, or asymmetric, ๐-player games where the strategy space of each player ๐ is endowed with a specified topology. This is because any ๐-player game โ can be symmetrized by letting the players switch roles, with all possible permutations considered. Thus, in the symmetric ๐-player game ๐ obtained by symmetrizing โ, the playersโ common 9 This notion of (topological) locality should not be confused with the one used in Section โ3, which refers to strategy profiles that differ only in the strategy of a single player. 10 This case corresponds to the choice of the trivial topology on ๐, where the only neighborhood of any strategy is the entire strategy set. 22 strategy space is the product space ๐ = ๐1 × ๐2 × โฏ × ๐๐ , whose elements are the strategy profiles in the original, asymmetric game and the topology is the product topology. For a player in ๐, a strategy ๐ฅ = (๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ ) โ ๐ determines the strategy ๐ฅ๐ the player will use when called to assume the role of any player ๐ in โ. All ๐! assignments of players in ๐ to roles in โ are considered, and a playerโs payoff is defined as his average payoff in these 1 ), 2 assignments. Formally, for any ๐ strategies in ๐, ๐ฅ 1 = (๐ฅ11 , ๐ฅ21 , โฆ , ๐ฅ๐ ๐ฅ = (๐ฅ12 , ๐ฅ22 , โฆ , ๐ฅ๐2 ), โฆ , ๐ฅ ๐ = (๐ฅ1๐ , ๐ฅ2๐ , โฆ , ๐ฅ๐๐ ), ๐(๐ฅ 1 , ๐ฅ 2 , โฆ , ๐ฅ ๐ ) = 1 ๐(1) ๐(2) ๐(๐) โ โ๐โ1 (1) (๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ ), ๐! (24) ๐โฮ where ฮ is the set of all permutation of (1,2, โฆ , ๐). (Note that superscripts in this formula index playersโ strategies in the symmetric game while subscripts refer to roles in the asymmetric one. For ๐ โ ฮ , ๐(๐) is the player assigned to role ๐.) A strategy profile ๐ฆ = (๐ฆ1 , ๐ฆ2 , โฆ , ๐ฆ๐ ) in โ may be viewed as stable, weakly stable or definitely unstable if it has the same property as a strategy in ๐. The following result expresses these conditions directly in terms of payoffs in the asymmetric game. Lemma 4. A strategy profile ๐ฆ in an ๐-player game โ is stable, weakly stable or definitely unstable as a strategy in the game obtained by symmetrizing โ if and only if it has a neighborhood where, for every strategy profile ๐ฅ โ ๐ฆ, the inequality โ 1 ๐ ๐ (|๐|) (โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โฬ ๐ โ ( ๐ฆ โฃ ๐ฅ๐ )) < 0, (25) a similar weak inequality or the reverse (strict) inequality, respectively, holds. The summation in (25) is over all sets of players ๐ โ {1,2, โฆ , ๐}. For each ๐, โฬ ๐ = (1/|๐|) โ๐โ๐ โ๐ (= 0, by convension, if ๐ = โ ) is the average payoff of the players in ๐, โฬ ๐ โ is the average for the complementary set ๐ โ , and ๐ฆ โฃ ๐ฅ๐ denotes the strategy profile where the players in and outside ๐ play according to ๐ฅ and ๐ฆ, respectively. Roughly, inequality (25) says that the ๐ฅ players fare worse on average than the ๐ฆ players. Proof. Since the game ๐ obtained by symmetrizing โ is invariant to permutations of its second through ๐th arguments, by elementary combinatorics the expression on the lefthand side of (23) is negative, positive or zero if and only if this is so for โ 1 ๐โ1 ๐ (|๐|โ1) (๐(๐ฅ, โโฆ , ๐ฅ , ๐ฆ, โโฆ , ๐ฅ )). โโฆ , ๐ฆ ) โ ๐(๐ฆ, โโฆ , ๐ฆ , ๐ฅ, 1โ๐ |๐| times ๐โ|๐| times (26) |๐| times ๐โ|๐| times The expression in parenthesis is the difference between the payoff of each player in ๐ when everyone in this set plays ๐ฅ and the players outside it play ๐ฆ and the payoff when the strategies are interchanged. Therefore, by (24), the sum (26) is equal to 1 1 (โ๐โ1 (1) ( ๐ฆ โฃ ๐ฅ๐โ1 (๐) ) โ โ๐โ1 (1) ( ๐ฅ โฃ ๐ฆ๐โ1 (๐) )). โโ ๐โ1 ๐! ) ๐ ( ๐โฮ 1โ๐ (27) |๐|โ1 The expression in parenthesis has the form โ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โ๐ ( ๐ฅ โฃ ๐ฆ๐ ), with ๐ โ ๐. Specifically, 23 ๐ = ๐โ1 (1) and ๐ = ๐โ1 (๐). For every pair (๐, ๐) with ๐ โ ๐ there are precisely (๐ โ 1)! pairs (๐, ๐) satisfying the last pair of equations (and, automatically, 1 โ ๐ and |๐| = |๐|). Therefore, (27) is equal to (๐ โ 1)! 1 1 (โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โฬ ๐ ( ๐ฅ โฃ ๐ฆ๐ )). โโ (โ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โ๐ ( ๐ฅ โฃ ๐ฆ๐ )) = โ ๐! ( ๐โ1 ) (๐) ๐โ โ ๐โ๐ ๐ |๐|โ1 |๐| Changing the summation variable in the expression โฬ ๐ ( ๐ฅ โฃ ๐ฆ๐ ) from ๐ to ๐ โ gives the lefthand side of (25). โ Symmetrization maps potential games to potential games. Indeed, the following lemma, whose easy proof is omitted, shows that, essentially, it also maps potentials to potentials. Lemma 5. If ๐ is a potential for an ๐-player โ, where the set of all strategy profiles is ๐, then a potential for the symmetric game obtained by symmetrizing โ is the symmetric function ๐น: ๐ ๐ โ โ defined by ๐น(๐ฅ 1 , ๐ฅ 2 , โฆ , ๐ฅ ๐ ) = 1 ๐(1) ๐(2) ๐(๐) โ ๐(๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ ). ๐! (28) ๐โฮ For potential games, stability and instability of strategy profiles have particularly simple characterizations in terms of the extremum points of the potential. Theorem 4. A strategy profile ๐ฆ in an ๐-player game โ with a potential ๐ is stable, weakly stable or definitely unstable as a strategy in the symmetric game ๐ obtained by symmetrizing โ if and only if ๐ฆ is a strict local maximum point, local maximum point or strict local minimum point of ๐, respectively. A global maximum point of ๐ is both globally weakly stable (and if it is a strict global maximum point, globally stable) in ๐ and an equilibrium in โ. Proof. Consider the potential ๐น for ๐ specified by (28). Since ๐น is symmetric (that is, invariant to permutations of its ๐ arguments), the left-hand site of (23) is equal to ๐ 1 1 โ(๐น(๐ฅ, โโฆ , ๐ฅ , ๐ฆ, โฆ , ๐ฆ) โ ๐น(๐ฅ, โโฆ , ๐ฅ , ๐ฆ, โฆ , ๐ฆ)) = (๐น(๐ฅ, ๐ฅ, โฆ ๐ฅ) โ ๐น(๐ฆ, ๐ฆ, โฆ ๐ฆ)). ๐ ๐ ๐ times ๐โ1 times ๐=1 Therefore, by Definition 3, ๐ฆ is stable, weakly stable or definitely unstable in ๐ if and only if it is a strict local maximum point, a local maximum point or a strict local minimum point, respectively, of the function ๐ฅ โฆ ๐น(๐ฅ, ๐ฅ, โฆ ๐ฅ). By (28), ๐น(๐ฅ, ๐ฅ, โฆ ๐ฅ) = ๐(๐ฅ), which implies that these conditions on ๐น are equivalent to those specified for ๐. The special case where the topology is the trivial one gives the last part of the theorem. โ A parallel result applies to strategies in population games. Theorem 5 (Milchtaich 2016). A strategy ๐ฆ in a population game ๐ with a potential ๐ท is stable, weakly stable or definitely unstable if and only if it is a strict local maximum point, local maximum point or strict local minimum point of ๐ท, respectively. In the first two cases, ๐ฆ is in addition an equilibrium strategy. If the potential ๐ท is strictly concave, a strategy is stable if and only if it is an equilibrium strategy. 24 Theorems 4 and 5 show that the equilibria considered in Theorem 1 are globally weakly stable in the relevant sense and the equilibria and equilibrium strategies in Theorems 2 and 3 are stable. The significance of these findings is elucidated by the next three theorems, which present fundamental links between static stability and the effect of altruism and spite on social welfare in general games. The first theorem is similar to Theorem 1 in that it concerns global comparative statics (Milchtaich 2012, Section 6), that is, comparison of two strategy profiles in two modified games corresponding to different altruism coefficients ๐ and ๐ , without assuming that the strategy profiles or the coefficients are close or that it is possible to connect them in a continuous manner. The other two theorems concern local comparative statics, which involve small, continuous changes to the altruism coefficient ๐ and the corresponding strategies or strategy profiles, and may be thought of as tracing the playersโ evolving behavior as they respond to the changing ๐. Note that, like Propositions 1 and 2, the three theorems hold for any choice of social payoff functions. Theorem 6. For an ๐-player game โ, a social payoff ๐, and altruism coefficients ๐ and ๐ with ๐ < ๐ โค 1, if two distinct strategy profiles ๐ฆ ๐ and ๐ฆ ๐ are globally weakly stable as strategies in the symmetric games obtained by symmetrizing the modified games โ๐ and โ ๐ , respectively, then ๐(๐ฆ ๐ ) โค ๐(๐ฆ ๐ ). If the strategies are globally stable, then a similar strict inequality holds. Proof. The proof uses the following identity, which holds for all (๐, ๐ and) ๐ฅ and ๐ฆ: (1 โ ๐ ) โ 1 ๐ ๐ (|๐|) (โฬ ๐๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โฬ ๐๐โ ( ๐ฆ โฃ ๐ฅ๐ )) + (1 โ ๐) โ 1 ๐ ๐ (|๐|) = (1 โ ๐ )(1 โ ๐) โ ๐ 1 ๐ (|๐| ) (โฬ ๐๐ ( ๐ฅ โฃ ๐ฆ๐ ) โ โฬ ๐๐ โ ( ๐ฅ โฃ ๐ฆ๐ )) ((โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ โฬ ๐ โ ( ๐ฆ โฃ ๐ฅ๐ )) + (โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ โ ) โ โฬ ๐ โ ( ๐ฆ โฃ ๐ฅ๐ โ ))) +(1 โ ๐ )๐โ(๐(๐ฅ) โ ๐(๐ฆ)) + (1 โ ๐)๐ โ(๐(๐ฆ) โ ๐(๐ฅ)) = (๐ โ ๐ )(๐(๐ฅ) โ ๐(๐ฆ)), where the second equality follows from the fact that, on its left-hand side, the coefficients of โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ ) and โฬ ๐ ( ๐ฆ โฃ ๐ฅ๐ โ ) vanish for every set of players ๐. By Lemma 4, applied to the trivial topology (see footnote 10), the two sums in the identity are nonpositive for ๐ฆ = ๐ฆ ๐ and ๐ฅ = ๐ฆ ๐ if these strategy profiles are globally weakly stable as strategies in the symmetric games obtained by symmetrizing the modified games โ๐ and โ ๐ , respectively, and they are negative if the strategies are globally stable. โ Theorem 7 (Milchtaich 2012). For an ๐-player game โ and a social payoff function ๐ that are both Borel measurable11, and altruism coefficients ๐0 and ๐1 with ๐0 < ๐1 โค 1, suppose that there is a continuous and finitely-many-to-one12 function assigning to each ๐0 โค ๐ โค ๐1 a strategy profile ๐ฆ ๐ such that the function ๐: [๐0 , ๐1 ] โ โ defined by 11 A sufficient condition for Borel measurability is that the function is continuous. 12 A function is finitely-many-to-one if the inverse image of every point is a finite set. 25 ๐(๐) = ๐(๐ฆ ๐ ) (29) is absolutely continuous.13 If, for every ๐0 < ๐ < ๐1 , the strategy profile ๐ฆ ๐ is stable, weakly stable or definitely unstable as a strategy in the game obtained by symmetrizing the modified game โ๐ , then ๐ is strictly increasing, nondecreasing or strictly decreasing, respectively. Theorem 8 (Milchtaich 2012). For a population game ๐ and a social payoff function ๐ that are both Borel measurable, and altruism coefficients ๐0 and ๐1 with ๐0 < ๐1 โค 1, suppose that there is a continuous and finitely-many-to-one function assigning to each ๐0 โค ๐ โค ๐1 a strategy ๐ฆ ๐ such that the function ๐: [๐0 , ๐1 ] โ โ defined by ๐(๐) = ๐(๐ฆ ๐ ) is absolutely continuous. If, for every ๐0 < ๐ < ๐1 , strategy ๐ฆ ๐ is a stable, weakly stable or definitely unstable in the modified game ๐๐ , then ๐ is strictly increasing, nondecreasing or strictly decreasing, respectively. These theorems present a broader perspective than that of Theorems 1, 2 and 3. First, they consider almost completely general games and social payoffs. Second, they are not restricted to equilibria or equilibrium strategies. Correspondingly, the two theorems concerning local comparative statics consider, not specific relations between altruism coefficients and strategies or strategy profiles, but any mapping ๐ โฆ ๐ฆ ๐ that is reasonably โsmoothโ (the continuity and absolute continuity conditions) and โresponsiveโ (the finitelymany-to-one condition). In this general setting, they show that social welfare necessary increases with increasing ๐ if each ๐ฆ ๐ is stable, and decreases in case of definite instability. As indicated, stability holds automatically for the strategy profiles and strategies considered in Theorems 2 and 3. Games with differentiable payoffs The next sufficient condition for stability may come in handy when stability is not automatically guaranteed. It is applicable to any ๐-player game โ where the strategy space of each player ๐ is a set in a Euclidean space โ๐๐ , for some (possibly, player-specific) ๐๐ โฅ 1. A strategy profile ๐ฅ = (๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ ) is viewed as an ๐-dimensional column vector, where ๐ = โ๐ ๐๐ . It is an interior strategy profile if each strategy ๐ฅ๐ lies in the interior of player ๐โs strategy space (equivalently, ๐ฅ lies in the interior of the product space). The gradient with respect to the components of player ๐โs strategy is denoted โ๐ and is viewed as an ๐๐ dimensional row vector (of first-order differential operators). For any ๐ and ๐, โT๐ โ๐ is therefore an ๐๐ × ๐๐ matrix (of second-order differential operators). For example, โT๐ โ๐ โ๐ is the Hessian matrix of player ๐โs payoff function with respect to the playerโs own strategy. These Hessian matrices are the diagonal blocks in the ๐ × ๐ block matrix โ1T โ1 โ1 ๐ป=( โฎ โT๐ โ1 โ๐ 13 โฏ โฑ โฏ โ1T โ๐ โ1 ). โฎ โT๐ โ๐ โ๐ A sufficient condition for absolute continuity is that the function is continuously differentiable. 26 (30) The value that the matrix ๐ป attains when its entries are evaluated at a strategy profile ๐ฅ is denoted ๐ป(๐ฅ). Proposition 7. In an ๐-player game โ where the strategy space of each player is a set in a Euclidean space, let ๐ฆ be an interior equilibrium that has a neighborhood where the playersโ payoff functions have continuous second-order derivatives. A sufficient condition for ๐ฆ to be stable or definitely unstable as a strategy in the game obtained by symmetrizing โ is that the symmetric matrix (1/2)(๐ป(๐ฆ) + ๐ป(๐ฆ)T ) is negative definite or positive definite, respectively.14 A necessary condition for weak stability is that the matrix is negative semidefinite. Proof. With ๐๐ denoting the characteristic function of a set of players ๐, the left-hand side of (25) can be written as 1 1 (๐๐ (๐)โโ๐ ( ๐ฆ โฃ ๐ฅ๐ ) โ ๐๐ โ (๐)โโ๐ ( ๐ฆ โฃ ๐ฅ๐ )). โโ ๐โ1 ๐ ๐ ๐ (|๐โ{๐}|) (31) For ๐ฅ tending to ๐ฆ, equivalently, ฯต๐ = ๐ฅ๐ โ ๐ฆ๐ โ 0 for all ๐, this expression can be written as 1 1 1 (๐๐ (๐) โ ๐๐ โ (๐)) (โ๐ + โ โ๐ โ๐ ฯต๐ + โ โ ฯตT๐ โT๐ โ๐ โ๐ ฯต๐ ) + ๐(โ๐โ2 ), โโ ๐โ1 ๐ 2 ๐ ๐ (|๐โ{๐}|) ๐โ๐ ๐โ๐ ๐โ๐ (32) where the payoff function โ๐ and its partial derivatives are evaluated at the point ๐ฆ and โ๐โ is the (Euclidean) length of the vector ฯต = (ฯต1 , ฯต2 , โฆ , ฯต๐ ) = ๐ฅ โ ๐ฆ. For each player ๐, the coefficient of โ๐ in (32) is 1 1 (๐๐ (๐) โ ๐๐ โ (๐)) โ ๐โ1 ๐ ( ) ๐ |๐โ{๐}| = 1 1 [(๐๐ (๐) โ ๐๐ โ (๐)) + (๐๐โช{๐} (๐) โ ๐(๐โช{๐})โ (๐))] = 0. โ ๐ ๐ ( ๐โ1 ) |๐โ{๐}| ๐โ๐ The last equality holds because the condition ๐ โ ๐ implies that the expression in square brackets is zero. For each ๐ and ๐, the coefficient of โ๐ โ๐ ฯต๐ in (32) is 1 1 (๐ (๐) โ ๐๐ โ (๐)), โ ๐ ๐ ( ๐โ1 ) ๐ |๐โ{๐}| ๐โ๐ which by a similar argument is zero if ๐ โ ๐. For each ๐, ๐ and ๐, the coefficient of ฯตT๐ โT๐ โ๐ โ๐ ฯต๐ is 1 1 (๐ (๐) โ ๐๐ โ (๐)), โ 2๐ ๐ ( ๐โ1 ) ๐ |๐โ{๐}| (33) ๐,๐โ๐ 14 The negative definiteness condition implies that ๐ป(๐ฆ) is D-stable. However, the second condition cannot be substituted for the first one in the proposition, as it does not imply even weak stability. 27 which again is zero if ๐ and ๐ are both different from ๐. If either ๐ โ ๐ and ๐ = ๐ or ๐ = ๐ and ๐ โ ๐, then (33) is equal to ๐ ๐ ๐=2 ๐=2 (๐โ2 ) 1 1 1 1 ๐โ1 1 ๐โ2 โ = โ ๐โ1 = โ = . 2๐ ๐ ( ๐โ1 ) 2๐ ๐โ1 4 ( ๐โ1 ) 2๐ ๐,๐โ๐ |๐|โ1 If ๐ = ๐ = ๐, then (33) is equal to ๐ (๐โ1 ) 1 1 1 1 ๐โ1 โ = โ ๐โ1 = . 2๐ ๐ ( ๐โ1 ) 2๐ ( ๐โ1 ) 2 ๐โ๐ ๐=1 |๐|โ1 Finally, โ๐ โ๐ = 0 for each player ๐, because the interior equilibrium ๐ฆ necessarily satisfies these first-order maximization conditions. Therefore, (32) reads (34) 1 1 โ ( โ ฯตT๐ โT๐ โ๐ โ๐ ฯต๐ + โ ฯตT๐ โT๐ โ๐ โ๐ ฯต๐ ) + ๐(โ๐โ2 ) 4 4 ๐ ๐ ๐ 1 1 = ฯตT (๐ป(๐ฆ) + ๐ป(๐ฆ)T )ฯต + ๐(โ๐โ2 ) = ฯตT ๐ป(๐ฆ)ฯต + ๐(โ๐โ2 ). 4 2 If (1/2)(๐ป(๐ฆ) + ๐ป(๐ฆ)T ) is negative definite or positive definite and ฯต โ 0, then ฯตT ๐ปฯต is negative or positive, respectively, and its absolute value is at least |๐0 |โ๐โ2 , where ๐0 โ 0 is the eigenvalue closest to 0 of (1/2)(๐ป(๐ฆ) + ๐ป(๐ฆ)T ). Therefore, in the first or second case, (34) is positive or negative for ๐ sufficiently close to 0, which proves that (25) or the reverse inequality, respectively, holds for ๐ฅ sufficiently close to ๐ฆ. If (1/2)(๐ป(๐ฆ) + ๐ป(๐ฆ)T ) is not negative semidefinite, then it has an eigenvector ๐ with eigenvalue ๐ > 0, so that ๐T ๐ป(๐ฆ)๐ is positive and equal to ๐โ๐โ2 . It follows that there are strategy profiles ๐ฅ arbitrarily close to ๐ฆ for which the reverse of (25) holds, which means that ๐ฆ is not weakly stable. โ Proposition 7 may be applied also to an interior equilibrium ๐ฆ ๐ in the modified game โ๐ , for any altruism coefficient ๐ and any social payoff function ๐ that has continuous second-order partial derivatives in a neighborhood of ๐ฆ ๐ . Here, the relevant matrix is ๐ป๐ , which is obtained from ๐ป be replacing โ with โ๐ . This observation suggests that, if the matrix (1/2)(๐ป๐ (๐ฆ ๐ ) + ๐ป๐ (๐ฆ ๐ )T ) is negative or positive definite, then it may be possible to use Theorem 7 to learn about the comparative statics of the game. Example 4. In a congestion game with splittable flow โ, there are ๐ โฅ 3 players and three resources. Player 1 has weight ๐ค1 = 1 and can only use resources ๐ and ๐ โฒ . Each of the other players ๐ has weight ๐ค๐ = 1/(๐ โ 1) and can only use resources ๐ and ๐ โณ . The (perunit) costs for ๐ โฒ and ๐ โณ are constant, ๐ โฒ and ๐ โณ respectively, and that of ๐ is given by a continuous cost function ๐๐ that is twice continuously differentiable outside the origin, with ๐๐โฒ (๐ก) > 0 and (๐๐ถ๐ )โฒ (๐ก) = 2๐๐โฒ (๐ก) + ๐กโ๐๐โณ (๐ก) โฅ 0, ๐ก > 0. (The marginal social cost ๐๐ถ๐ is defined in (22).) Cominetti et al. (2009) use this setting 28 (35) for deriving lower bounds on the price of anarchy with respect to the aggregate payoff (or cost) for the power function ๐๐ (๐ก) = ๐ก ๐ . (36) For example, with ๐ = 1, ๐ โฒ = 2.8 and ๐ โณ = 1.9, they show that if the number of players ๐ is large, then the game has a unique equilibrium, in which all players use both their resources, and the price of anarchy is close to 1.341. This raises the question of how the aggregate cost at equilibrium would be affected by increasing the altruism coefficient from 0 to some (small) positive value ๐. Claim. Suppose that the cost function ๐๐ satisfies ๐กโ๐๐โณ (๐ก) ๐+1 , โฒ (๐ก) < 2 ๐๐ ๐โ1 ๐ก > 0. If โ has an equilibrium where all players use both their resources, then the modified game โ๐ has a unique such equilibrium ๐ฆ ๐ for all ๐ in some interval [0, ๐1 ], with 0 < ๐1 < 1, and the corresponding aggregate payoff ๐(๐ฆ ๐ ) is strictly increasing (and the aggregate cost is strictly decreasing) in this interval. For the power function (36), the above condition on ๐๐ (together with (35)) translates to 0<๐< 3๐ + 1 . ๐โ1 This is the same condition obtained, in a more general setting, by Altman el at. (2002) as a sufficient condition for the uniqueness of the equilibrium in the unmodified game โ.15 This coincidence is not surprising, as the sufficient condition for uniqueness employed by these authors is the negative definiteness of a matrix similar to (1/2)(๐ป + ๐ปT ) (or the positive definiteness of its negative). Note that, with ๐๐ given by a power function, โ is a potential game only in the linear case, ๐ = 1. To prove the Claim, note first that, since each player ๐ can only use two resources, the playerโs strategy ๐ฅ๐ is completely specified by the component ๐ฅ๐๐ . It is therefore possible to identify ๐ฅ๐ with ๐ฅ๐๐ , so that the strategy space becomes unidimensional, specifically, the interval [0, ๐ค๐ ]. Thus, a strategy profile has the form ๐ฅ = (๐ฅ1๐ , ๐ฅ2๐ , โฆ , ๐ฅ๐๐ ). For 0 โค ๐ โค 1, the derivative of player 1โs payoff in the modified game โ๐ with respect to the playerโs strategy, denoted (โ1๐ )1 , at a point ๐ฅ โ 0 is given by โ(โ1๐ )1 (๐ฅ) = ๐๐ (๐ฅ๐ ) + ((1 โ ๐)๐ฅ1๐ + ๐๐ฅ๐ )๐๐โฒ (๐ฅ๐ ) โ ๐ โฒ , and the second derivative is given by 15 Because of a typo in that paper, the condition obtained there is presented in a slightly altered, stronger form (Nahum Shimkin, personal communication). 29 โ(โ1๐ )11 (๐ฅ) = 2๐๐โฒ (๐ฅ๐ ) + ((1 โ ๐)๐ฅ1๐ + ๐๐ฅ๐ )๐๐โณ (๐ฅ๐ ) = 2(1 โ ๐) (1 โ ๐ฅ1๐ โฒ ๐ฅ1๐ ) ๐๐ (๐ฅ๐ ) + ((1 โ ๐) + ๐) (๐๐ถ๐ )โฒ (๐ฅ๐ ) ๐ฅ๐ ๐ฅ๐ โฅ 0. The inequality, which follows from (35), shows that the payoff is concave with respect to player 1โs own strategy ๐ฅ1๐ . The same conclusion holds for any other player ๐, as the only difference is that ๐ โฒ in this case is replaced with ๐ โณ . Therefore, an interior equilibrium ๐ฆ ๐ in โ๐ is completely characterized by the first-order conditions (โ๐๐ )๐ (๐ฆ ๐ ) = 0 (๐ = 1,2, โฆ , ๐). Summation of these equations shows that ๐ฆ ๐ must be a solution of ๐๐๐ (๐ฅ๐ ) + (1 + ๐(๐ โ 1))โ๐ฅ๐ ๐๐โฒ (๐ฅ๐ ) = ๐ โฒ + (๐ โ 1)๐ โณ . (37) The expression on the left-hand side can be written as (1 โ ๐)(๐ โ 1)โ๐๐ (๐ฅ๐ ) + (1 + ๐(๐ โ 1))โ๐๐ถ๐ (๐ฅ๐ ). By (35), for 0 โค ๐ < 1, the derivative of this expression with respect to ๐ฅ๐ and the derivative with respect to ๐ are both positive if ๐ฅ๐ > 0. Therefore, for such ๐, a positive solution to (37) is unique and, by the implicit function theorem, is determined by ๐ as a continuously differentiable function with a negative derivative. Thus, if ๐ฆ 0 is an interior equilibrium in the unmodified game โ, there exists some 0 < ๐1 < 1 and a continuously differentiable mapping ๐ โฆ ๐ฆ๐๐ on [0, ๐1 ] with ๐๐ฆ๐๐ / ๐๐ < 0 such that ๐ฆ๐๐ solves (37). Setting ๐ฅ๐ = ๐ฆ๐๐ in each of the first-order conditions (โ๐๐ )๐ = 0 now gives a linear equation for ๐ฅ๐๐ . If ๐1 is ๐ ๐ sufficiently close to 0, the solution ๐ฆ๐๐ satisfies 0 < ๐ฆ๐๐ < ๐ค๐ for every 0 โค ๐ โค ๐1 , and the ๐ ๐ ๐ ๐ derivative ๐๐ฆ๐๐ / ๐๐ exists and is continuous. (Necessarily, ๐ฆ2๐ = ๐ฆ3๐ = โฏ = ๐ฆ๐๐ , as the corresponding ๐ โ 1 equations are identical.) Summation of the ๐ equations and ๐ comparison with (37) show that ๐ฆ๐๐ necessarily sum up to ๐ฆ๐๐ , so that they specify the unique interior equilibrium in โ๐ . In addition, since ๐๐ฆ๐๐ / ๐๐ < 0, the function ๐ โฆ ๐ฆ ๐ is one-to-one on [0, ๐1 ] and ๐๐ฆ ๐ / ๐๐ โ 0 there. By the last conclusion, Proposition 7 and Theorem 7, a sufficient condition for ๐ โฆ ๐ฆ ๐ to be strictly increasing in [0, ๐1 ] is that the matrix (1/2)(๐ป๐ (๐ฆ ๐ ) + ๐ป๐ (๐ฆ ๐ )T ) is positive definite ๐ for 0 < ๐ < ๐1 . The matrix ๐ป๐ = ((โ๐๐ )๐๐ ) ๐,๐=1 is explicitly given by the following equations. For all ๐ and ๐ โ ๐, โ(โ๐๐ )๐๐ (๐ฅ) = 2๐๐โฒ (๐ฅ๐ ) + ((1 โ ๐)๐ฅ๐๐ + ๐๐ฅ๐ )๐๐โณ (๐ฅ๐ ), โ(โ๐๐ )๐๐ (๐ฅ) = (1 + ๐)๐๐โฒ (๐ฅ๐ ) + ((1 โ ๐)๐ฅ๐๐ + ๐๐ฅ๐ ) ๐๐โณ (๐ฅ๐ ). Therefore, โ(1/2)(๐ป๐ (๐ฆ ๐ ) + ๐ป๐ (๐ฆ ๐ )T ) = (1 โ ๐) (๐ผ๐ผ + ๐ฝ๐ธ + ๐พ(๐น + ๐น T )) + ๐๐ฟ๐ธ, (38) where ๐ผ is the identity matrix, ๐ธ is the matrix with all entries 1, ๐น has 1โs in the first row and 0โs elsewhere, 30 ๐ผ ๐ฝ ๐พ ๐ฟ = ๐๐โฒ (๐ฆ๐๐ ) ๐ โณ (๐ฆ ๐ ) = ๐๐โฒ (๐ฆ๐๐ ) + ๐ฆ2๐ ๐๐ ๐ ๐ ๐ )๐ โณ (๐ฆ ๐ ) = (1/2)(๐ฆ1๐ โ ๐ฆ2๐ ๐ ๐ = 2๐๐โฒ (๐ฆ๐๐ ) + ๐ฆ๐๐ ๐๐โณ (๐ฆ๐๐ ). ๐ ๐ )/(๐ Note that, since ๐ฆ2๐ = (๐ฆ๐๐ โ ๐ฆ1๐ โ 1) < ๐ฆ๐๐ /2, it follows from (35) that the expressions ๐ผ, ๐ฝ, ๐ฝ + ๐พ and ๐ฟ are positive. If ๐พ = 0, then, since ๐ธ is positive semidefinite, the right-hand side of (38) is positive definite. If ๐พ โ 0, then to reach the same conclusion it suffices to show that all the eigenvalues of the matrix ๐ฝ๐ธ + ๐พ(๐น + ๐น T ) are greater than โ๐ผ. The rank of this ๐ × ๐ matrix is 2. Its two non-zero eigenvalues are the roots of the quadratic polynomial ๐(๐ ) = ๐ 2 โ (๐๐ฝ + 2๐พ)๐ โ (๐ โ 1)๐พ 2 , because it is not difficult to check that the condition for ((๐ โ (๐ โ 1)๐ฝ)/(๐ฝ + ๐พ), 1,1, โฆ ,1) to be an eigenvector with eigenvalue ๐ is ๐(๐ ) = 0. Since ๐(0) < 0, the negative number โ๐ผ is smaller than both roots of ๐ if and only if ๐(โ๐ผ) > 0. Since ๐๐ฝ + 2๐พ = ๐๐๐โฒ (๐ฆ๐๐ ) + ๐ฆ๐๐ ๐๐โณ (๐ฆ๐๐ ) and |๐พ| โค (1/2)|๐ฆ๐๐ ๐๐โณ (๐ฆ๐๐ )|, a sufficient condition for ๐(โ๐ผ) > 0 is that, for all ๐ก > 0, 2 2 (๐๐โฒ (๐ก)) โ (๐๐๐โฒ (๐ก) + ๐ก๐๐โณ (๐ก))(โ๐๐โฒ (๐ก)) โ (1/4)(๐ โ 1)(๐ก๐๐โณ (๐ก)) > 0. The solution of this quadratic inequality is โ2 < ๐ก๐๐โณ (๐ก) ๐+1 . โฒ (๐ก) < 2 ๐๐ ๐โ1 As the left inequality holds by (35), this completes the proof of the Claim. As an alternative to the use of Proposition 7 and Theorem 7 in the proof of the Claim, the following proposition could be used. The proposition directly links the definiteness of the matrix (1/2)(๐ป๐ (๐ฆ ๐ ) + ๐ป๐ (๐ฆ ๐ )T ) (equivalently, of the quadratic form corresponding to ๐ป๐ (๐ฆ ๐ )) with the effect of altruism and spite on the social payoff. Proposition 8. For an ๐-player game โ where the strategy space of each player is a set in a Euclidean space, a social payoff function ๐, and altruism coefficients ๐0 and ๐1 with ๐0 < ๐1 , suppose that there is a continuously differentiable function assigning to each ๐0 < ๐ < ๐1 an interior equilibrium ๐ฆ ๐ in the modified game โ๐ that has a neighborhood where the playersโ payoff functions and ๐ have continuous second-order derivatives. At each point ๐0 < ๐ < ๐1 , T ๐ ๐๐ฆ ๐ ๐๐ฆ ๐ ๐(๐ฆ ๐ ) = โ(1 โ ๐) ( ) ๐ป๐ (๐ฆ ๐ )โ. ๐๐ ๐๐ ๐๐ Proof. Since ๐ฆ ๐ is an interior equilibrium in โ๐ , it satisfies the necessary first-order maximization condition for each player ๐, which is โ๐ โ๐๐ (๐ฆ ๐ ) = โ๐ ((1 โ ๐)โ๐ + ๐๐)(๐ฆ ๐ ) = 0. Total derivation with respect to ๐ gives 31 (39) ๐ T ๐๐ฆ๐๐ ๐ โ๐ โ๐๐ (๐ฆ ๐ ) = โ๐ (โโ๐ + ๐)(๐ฆ ๐ ) + โ ( ) โ๐T โ๐ โ๐๐ (๐ฆ ๐ ) = 0. ๐๐ ๐๐ ๐=1 Post-multiplication of both sides of the right equality with (1 โ ๐) ๐๐ฆ๐๐ / ๐๐ and summation over ๐ give ๐ โ โ๐ (โ(1 โ ๐)โ๐ + (1 โ ๐)๐)(๐ฆ T ๐) ๐=1 ๐ ๐๐ฆ๐๐ ๐๐ฆ ๐ T ๐๐ฆ + (1 โ ๐) ( ) (๐ป๐ (๐ฆ ๐ )) = 0. ๐๐ ๐๐ ๐๐ By the second equality in (39), the first term above (the sum) is equal to ๐๐(๐ฆ ๐ )/ ๐๐. โ 7 Summary and Discussion Users of common resources exert negatives externalities on the other users, and are similarly affected by them. It therefore seems logical that any degree of altruism, which expresses internalization of other peopleโs welfare, would make everyone better off. As this paper shows, this is not always so. The analysis presented here only concerns the expected material consequences (hence, the social desirability) of internalization of social welfare, that is, its effect on the actual level of social welfare at equilibrium. The questions of the origin of such attitudes and their prevalence lie outside the scope of the present work. In some contexts, altruism may have a biological basis. If the individuals involved are related, then the effect of the use of a common resource on the inclusive fitness of individual ๐ is measured by the effect on ๐โs own fitness plus ๐ times that of each of the other individuals ๐, where ๐ is the coefficient of relatedness between ๐ and ๐ (Milchtaich 2006a). In a social context, a similar parameter ๐ may express individual ๐โs level of empathy towards ๐. Edgeworth (1881) calls this parameter the coefficient of effective sympathy. More generally, the altruism coefficient ๐ may specify the weight a person attaches to some completely general social payoff, like the degree of global warming, that is not necessarily expressible in terms of the individualsโ personal, material payoffs but is a function of everyoneโs actions. Some of the results in this paper apply to such general social payoff functions while others are specific to a particular function, the aggregate personal payoff (or the negative of the aggregate cost). In addition, some results also concern negative ๐, which may be interpreted as spite. A strong simplifying assumption made in this paper is that all the individuals involved are equally altruistic: they have the same ๐. One interpretation of such a common altruism coefficient is that it represents common values that are shared by all users of the common resources. Another assumption, which is implicit in the functional form of the modified payoff, is that the latter is affected linearly by the social payoff.16 A more complicated effect would require adding more terms to the function. However, these terms may be expected to 16 The modified payoff is linear also in ๐. However, this linearity does not represent a strong assumption because multiplying the expression on the right-hand side of (1) by any positive-valued function of ๐ would not affect the preferences expressed by it. In other words, only the ratio (1 โ ๐): ๐, which gives the marginal rate of substitution of personal payoff for social payoff, matters. 32 be relatively unimportant if the effect is weak, that is, the personal payoff predominates. In this case, a linear relation with a small ๐ may be a good approximation. Many of the phenomena described in this paper are evident already for arbitrarily small ๐. The simplest scenario where the use of common resources gives rise to negative externalities is that of an unweighted congestion game. As shown, it is generally not true in this setting that altruism always increases the aggregate personal payoff. In fact, even with linear cost functions, inefficient equilibria may exist only for altruism coefficients close to 1, which reflect complete or almost complete selflessness. In other words, as the altruism coefficient ๐ increases, the price of anarchy in the game may (weakly) increase. The main results in Section โ3 concern the characterization of the games for which this is never so. A sufficient condition for nonincreasing price of anarchy is the monotonicity property, which guarantees that only strategy profiles that maximize the gameโs potential function are equilibria. The monotonicity property only concerns the โcongestion game formโ; the cost functions are irrelevant. For network congestion games, this fact translates to a condition on the topology of the undirected network on which the game is defined. In particular, if that network is extension-parallel, then the monotonicity property automatically holds for the game, so that altruism can only increase social welfare or leave it unchanged. The same it not true for general series-parallel networks. Allowing users to split their demand among several strategies has a significant effect on comparative statics. In particular, with linear, increasing costs, the aggregate personal payoff at equilibrium is uniquely determined by the altruism coefficient ๐ and is nondecreasing in [โ1,1]. That is, an increasingly positive or negative ๐ can only increase or decrease social welfare, respectively, or leave it unchanged. The highest aggregate payoff is attained with complete selflessness, ๐ = 1. Internalization of social payoff may occur even if a single personโs ability to affect the social payoff is practically nonexistent. For example, people may care about their carbon footprint even while recognizing its insignificance on a global scale. Their attitude may be justified by the assertion that if many people acted similarly, the effect would not be insignificant. Put differently, the emphasis is on the marginal effect of the act on the social payoff, that is, the effect per unit mass of actors. For example, in the special case where the total effect is proportion to the size of the set of actors, the marginal effect of an action expresses the outcome if everybody switched to that action. In this context, the altruism coefficient ๐ is the weight attached to the marginal effect, and comparative statics concern the relation between this weight and the actual level of social payoff at equilibrium. For congestion games, the relevant model is that of a population game arising from a nonatomic congestion game, where the set of players is modeled as a continuum. The social payoff is the mean payoff, which is analogous to the aggregate personal payoff in games with a finite number of players. As shown, comparative statics are qualitatively quite similar to the case of a finite number of users with splittable flow. For ๐ between 0 and 1, the mean payoff at equilibrium is uniquely determined by the altruism coefficient as a nondecreasing function, which reaches its maximum at ๐ = 1. 33 The results presented above rely to a large extent on the fact that the games considered are potential games and the equilibria involved are necessarily maximizers of the potential. However, Section โ6 points to a deeper link underlying the connection between comparative statics and the potential function. Specifically, it shows that a strict maximum point of the potential is necessarily (statically) stable, and that stability or definite instability, respectively, necessarily brings about โpositiveโ or โnegativeโ comparative statics, in the sense of the effect of increasing altruism of the social payoff. 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Marginal cost pricing for atomic network congestion games. Technical report. 35 Appendix A directed or undirected two-terminal network is, respectively, a directed or undirected multigraph with terminal vertices ๐ and ๐ that satisfies the โirredundancyโ condition: each edge ๐ and each vertex ๐ฃ is included in at least one route in the network. In the undirected case, a route is defined as a (simple) path of the form ๐๐1 ๐ฃ1 โฏ ๐ฃ๐โ1 ๐๐ ๐ (๐ โฅ 1), that is, one that begins at the origin vertex ๐ and ends at the destination vertex ๐. (An alternative, simpler, notation is ๐1 ๐2 โฏ ๐๐ . ) In the directed case, a route is also required to traverse each of its edges ๐ in the direction assigned to ๐, that is, the tail and head vertices of ๐ must be its immediate predecessor and successor, respectively, in the route. This difference means that the set of routes in a directed two-terminal network is a subset of the set of routes in the networkโs undirected version, which is obtained by simply ignoring the edge directions. An arbitrary assignment of directions to the edges of an undirected two-terminal network ๐บ does not necessarily give a directed network, as the irredundancy condition may fail to hold. If the irredundancy condition does hold, the resulting directed two-terminal network is said to be a directed version of ๐บ. Proposition A1. Every undirected two-terminal network has at least one directed version. Proof. The proof is by induction on the number of edges in the network. If there is only one edge, the assertion is trivial. To establish the inductive step, consider a network ๐บ with more than one edge, and some edge ๐ incident with the origin ๐. If no other edge is incident with ๐, then by the induction hypothesis there exists at least one directed version of the network obtained from ๐บ by contracting ๐, that is, eliminating the edge and its non-terminal vertex ๐ฃ and replacing ๐ฃ with ๐ as the terminal vertex of all the edges originally incident with ๐ฃ. Any such directed version gives a directed version of ๐บ, in which ๐ is directed from ๐ to ๐ฃ. Suppose, then, that ๐ is not the only edge incident with ๐, and consider the subnetwork ๐บ๐ of ๐บ whose edges and vertices are all those that belong to some route in ๐บ where the first edge is ๐. Claim. If a route ๐ in ๐บ includes a non-terminal vertex ๐ข that is in ๐บ๐ , then every edge (hence, also every vertex) that follows ๐ข in ๐ is also in ๐บ๐ . It clearly suffices to establish this for the edge ๐ โฒ that immediately follows ๐ข in ๐. Suppose that ๐ โฒ is not in ๐บ๐ . Let ๐ฃ the first vertex that follows ๐ โฒ in ๐ and is in ๐บ๐ . All the edges and vertices in ๐ between ๐ข and ๐ฃ are not in ๐บ๐ . Adding them to it creates a new subnetwork of ๐บ, since the addition is equivalent to adding to ๐บ๐ a single edge with end vertices ๐ข and ๐ฃ and then subdividing the edge, if needed, one or more times (Milchtaich 2015, Section 2.2). By definition, there exists in the new subnetwork a route ๐ โฒ that includes ๐ โฒ . Necessarily, ๐ โฒ also includes ๐. The conclusion contradicts the assumption that ๐ โฒ is not in ๐บ๐ , and thus proves the claim. Consider now the collections of all edges and vertices that are obtained from those in ๐บ by (i) eliminating all edges that are in ๐บ๐ and (ii) identifying all non-terminal vertices that are in ๐บ๐ with ๐. It follows from the Claim that these collections constitute a network ๐บ โฒ (with the terminal vertices ๐ and ๐). To see this, consider any edge or non-terminal vertex in them and a route ๐ in ๐บ that includes it. It follows from the Claim that the inclusion still holds if ๐ is 36 replaced with its section ๐๐๐ข , which is the path obtained from ๐ by deleting all edges and vertices that follow ๐ข, where ๐ข is the first vertex in ๐, other than ๐, that is in ๐บ๐ . As ๐ข is one of the vertices identified with ๐, ๐๐๐ข is a route in ๐บ โฒ . By the induction hypothesis, both ๐บ๐ and ๐บ โฒ have directed versions, which together specify directions for all edges in ๐บ. It remains to show that these directions define a directed version of ๐บ, that is, the irredundancy condition holds. Any edge or vertex that is or is not in ๐บ๐ is included in some route ๐ in the directed version of ๐บ๐ or some route ๐ โฒ in the directed version of ๐บ โฒ , respectively. By construction, such routes honor the edgesโ directions in ๐บ. Route ๐ is automatically a route also in ๐บ, whereas ๐ โฒ is a path in ๐บ that starts at ๐, ends at some vertex ๐ข in ๐บ๐ , and does not include any other vertex or edge that is in ๐บ๐ . It is therefore possible to extend ๐ โฒ to a route in ๐บ that honors the edgesโ directions by replacing โณ ๐ข with the section ๐๐ข๐ of some route ๐ โณ in the directed version of ๐บ๐ that includes ๐ข, where โณ ๐๐ข๐ is obtained from ๐ โณ by deleting all the edges and vertices that precede ๐ข. โ Corollary A1. The collection of the undirected versions of all directed two-terminal networks coincides with the collection of all undirected two-terminal networks. Clearly, an assignment of directions to the edges of an undirected two-terminal network ๐บ defines a directed version of ๐บ only if it satisfies the condition mentioned in Section โ3.1, which is that the direction of each edge ๐ coincides with the direction in which some route in (the undirected network) ๐บ traverses ๐. However, this necessary condition is not sufficient. For example, an assignment satisfying it may render a non-terminal degree-two vertex ๐ฃ the head vertex of both edges incident with ๐ฃ, which means that the irredundancy condition for directed networks is violated. However, it follows from Proposition A1 that if there is only one assignment of directions that satisfies the above condition, then that assignment necessarily does give a directed version of ๐บ. Such uniqueness is rather special. Indeed, the following result can easily be deduced from Proposition 1 in Milchtaich (2006b). Corollary A2. For an undirected two-terminal network ๐บ, the following conditions are equivalent: (i) (ii) (iii) For each edge ๐, all routes in ๐บ that include ๐ traverse it in the same direction. For every assignment of allowable directions to the edges in ๐บ such that the direction of each edge ๐ is that in which some route in ๐บ traverses ๐, all routes in ๐บ are allowable. ๐บ is series-parallel. 37