Download Internalization of Social Cost in Congestion Games

Internalization of Social Cost in
Congestion Games
Igal Milchtaich*
December 2016
Congestion models may be studied from either the users’ 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 𝑟 that is
not necessarily 1 or 0, and may also be negative. It examines the comparative-statics
question of whether higher 𝑟 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 𝑟 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 𝑟 to each of the other players’ payoff, the result may
paradoxically be lower personal payoffs than in the case 𝑟 = 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 𝑟, 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 𝑟 (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 “negative”
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 players’ personal costs, which is unaffected by the altruism coefficient 𝑟. Yet the price of
anarchy is affected, because the set of equilibria may chance as 𝑟 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 𝑟, 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 𝑟 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 𝑟. 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
“positive”, or “normal” 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 “bad” 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 ℎ specifies a (finite or infinite) set of players and, for each player 𝑖, a
strategy set 𝑋𝑖 and a payoff function ℎ𝑖 : 𝑋 → ℝ, where 𝑋 = ∏𝑖 𝑋𝑖 is the set of all strategy
profiles. A social payoff is any function 𝑓: 𝑋 → ℝ. An important example in a game with
finitely many players is the aggregate payoff, 𝑓 = ∑𝑖 ℎ𝑖 . The altruism coefficient is an
exogenously given parameter 𝑟 ≤ 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 ℎ𝑟 , where the payoff of each player 𝑖 is not the original, personal
payoff ℎ𝑖 but the modified payoff
ℎ𝑖𝑟 = (1 − 𝑟)ℎ𝑖 + 𝑟𝑓.
(1)
Comparative statics concern the connection between 𝑟 and the value of 𝑓 at the Nash
equilibria of the modified game ℎ𝑟 . In the special case where 𝑓 is the aggregate payoff, so
that
ℎ𝑖𝑟 = ℎ𝑖 + 𝑟 ∑ ℎ𝑗 ,
𝑗≠𝑖
this connection expresses the effect on the actual equilibrium aggregate personal payoff
when each of the players attaches the same weight 𝑟 to the personal payoff of each of the
other players. A positive, negative or zero 𝑟 may be interpreted as expressing altruism, spite
or complete selfishness, respectively.
A symmetric 𝑁-player game is one where all players share the same strategy set 𝑋 and their
payoffs are specified by a single payoff function 𝑔: 𝑋 𝑁 → ℝ that is invariant to permutations
of its second through 𝑁th arguments. If one player uses strategy 𝑥 and the others use
𝑦, 𝑧, … , 𝑤, in any order, the first player’s payoff is 𝑔(𝑥, 𝑦, 𝑧, … , 𝑤).
A population game is formally defined as a symmetric two-player game 𝑔 where the strategy
space2 𝑋 is a convex set in a (Hausdorff real) linear topological space (for example, the unit
simplex in a Euclidean space ℝ𝑛 ) and 𝑔(𝑥, 𝑦) is continuous in the second argument 𝑦 for all
𝑥 ∈ 𝑋 (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 “playing the field”. This means that an individual’s payoff 𝑔(𝑥, 𝑦)
depends only on his own strategy 𝑥 and on a suitability defined population strategy 𝑦 ∈ 𝑋,
which encapsulates the other players’ actions.
A social payoff for a population game is any continuous function 𝜙: 𝑋̂ → ℝ whose domain 𝑋̂
is the cone of the strategy space 𝑋. The cone consists of all elements of the form 𝑡𝑥, 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
𝑥 ∈ 𝑋 and 𝑡 > 0. Note that, unlike for 𝑁-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 𝑔 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 one’s
actions on the social payoff 𝜙 (Chen and Kempe 2008). To formalize this idea, suppose that
𝜙 has a directional derivative in every direction 𝑥̂ ∈ 𝑋̂ and that the derivative depends
continuously on the point 𝑦̂ ∈ 𝑋̂ at which it is evaluated. In other words, the assumption is
that the differential 𝑑𝜙: 𝑋̂ 2 → ℝ, defined by (the one-sided derivative)
𝑑𝜙(𝑥̂, 𝑦̂) =
𝑑
|
𝜙(𝑡𝑥̂ + 𝑦̂),
𝑑𝑡 𝑡=0+
exists and is continuous in the second argument (Milchtaich 2012). For altruism coefficient
𝑟 ≤ 1, the modified game 𝑔𝑟 is then defined by
𝑔𝑟 (𝑥, 𝑦) = (1 − 𝑟)𝑔(𝑥, 𝑦) + 𝑟 𝑑𝜙(𝑥, 𝑦).
(2)
Comparative statics concern the connection between the altruism coefficient 𝑟 and the
value of the social payoff 𝜙 at the equilibrium strategies in 𝑔𝑟 , that is, those (population)
strategies 𝑦 ∈ 𝑋 satisfying
𝑔𝑟 (𝑦, 𝑦) ≥ 𝑔𝑟 (𝑥, 𝑦),
𝑥 ∈ 𝑋.
(3)
2.1 Potential games
A game ℎ 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 𝑃: 𝑋 → ℝ such that, whenever a
single player 𝑖 changes his strategy, the resulting change in 𝑖’s payoff equals the change in 𝑃:
ℎ𝑖 ( 𝑦 ∣ 𝑥𝑖 ) − ℎ𝑖 (𝑦) = 𝑃( 𝑦 ∣ 𝑥𝑖 ) − 𝑃(𝑦),
𝑥𝑖 ∈ 𝑋𝑖 , 𝑦 ∈ 𝑋.
Here, 𝑋𝑖 is player 𝑖’s strategy set, 𝑋 is the set of all strategy profiles, and 𝑦 ∣ 𝑥𝑖 denotes the
strategy profile that differs from 𝑦 only in that player 𝑖 uses strategy 𝑥𝑖 .
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 𝛷: 𝑋 → ℝ is an (exact) potential for a population game 𝑔
with a strategy space 𝑋 if, for all 𝑥, 𝑦 ∈ 𝑋 and 0 < 𝑝 < 1, the derivative on the left-hand side
of the following equality exists and the equality holds:
𝑑
𝛷(𝑝𝑥 + (1 − 𝑝)𝑦) = 𝑔(𝑥, 𝑝𝑥 + (1 − 𝑝)𝑦) − 𝑔(𝑦, 𝑝𝑥 + (1 − 𝑝)𝑦).
𝑑𝑝
(4)
A useful fact (Milchtaich 2016) is that a sufficient condition for a function 𝛷: 𝑋 → ℝ to be a
potential for 𝑔 is that it can be extended to a continuous function on 𝑋̂, the cone of 𝑋, such
that for the extended function (which is also denoted by 𝛷) the differential 𝑑𝛷: 𝑋̂ 2 → ℝ
exists, is continuous in the second argument and satisfies
4
𝑑𝛷(𝑥, 𝑦) = 𝑔(𝑥, 𝑦),
𝑥, 𝑦 ∈ 𝑋.
(5)
For a game ℎ with a potential 𝑃, and any social payoff 𝑓 and altruism coefficient 𝑟, the
modified game ℎ𝑟 is also a potential game, as it is easy to see that the function 𝑃𝑟 defined as
follows is a potential for it:
𝑃𝑟 = (1 − 𝑟)𝑃 + 𝑟𝑓.
(6)
Similarly, for a population game 𝑔 with a potential 𝛷, and any social payoff 𝜙 and
altruism coefficient 𝑟, the modified game 𝑔𝑟 is also a potential game, as the following
function 𝛷 𝑟 is a potential:3
𝛷 𝑟 = (1 − 𝑟)𝛷 + 𝑟𝜙.
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 ℎ with a potential 𝑃, a social payoff 𝑓, and altruism coefficients 𝑟
and 𝑠 with 𝑟 < 𝑠 ≤ 1, if two strategy profiles 𝑦 𝑟 and 𝑦 𝑠 are (global) maximum points of 𝑃𝑟
and 𝑃 𝑠 , respectively, then
𝑓(𝑦 𝑟 ) ≤ 𝑓(𝑦 𝑠 ).
If 𝑦 𝑟 and 𝑦 𝑠 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):
(𝑟 − 𝑠)(𝑓(𝑥) − 𝑓(𝑦)) = (1 − 𝑠)(𝑃𝑟 (𝑥) − 𝑃𝑟 (𝑦)) + (1 − 𝑟)(𝑃 𝑠 (𝑦) − 𝑃 𝑠 (𝑥)).
The equality shows that the difference 𝑓(𝑥) − 𝑓(𝑦) has the opposite sign to that of the
expression on the right-hand side. The latter is nonpositive for 𝑦 = 𝑦 𝑟 and 𝑥 = 𝑦 𝑠 if these
strategy profiles maximize 𝑃𝑟 and 𝑃 𝑠 , respectively, and it is negative if they are strict
maximum points.
∎
Proposition 2. For a population game 𝑔 with a potential 𝛷, a social payoff 𝜙, and altruism
coefficients 𝑟 and 𝑠 with 𝑟 < 𝑠 ≤ 1, if two strategies 𝑦 𝑟 and 𝑦 𝑠 are maximum points of 𝛷 𝑟
and 𝛷 𝑠 , respectively, then
𝜙(𝑦 𝑟 ) ≤ 𝜙(𝑦 𝑠 ).
If 𝑦 𝑟 and 𝑦 𝑠 are strict maximum points, then a similar strict inequality holds.
In a game ℎ with a potential 𝑃, a global maximum point of 𝑃 is clearly an equilibrium. In a
population game 𝑔 with a potential 𝛷, even a local maximum point of 𝛷 is an equilibrium
strategy, and if 𝛷 is a convex function, then the converse also holds, in fact, an equilibrium
By definition of 𝑔1 , a condition similar to (5) holds with 𝛷 and 𝑔 replaced by 𝜙 and 𝑔1 , respectively.
This proves that 𝜙 (more precisely, its restriction to 𝑋) is a potential for 𝑔1 (= 𝑑𝜙), which implies that
𝛷𝑟 is a potential for 𝑔𝑟 .
3
5
strategy is necessarily a global maximum point of 𝛷. These assertions follow as immediate
conclusions from the following identity, which is a corollary of (4):
𝑑
|
𝛷(𝑝𝑥 + (1 − 𝑝)𝑦) = 𝑔(𝑥, 𝑦) − 𝑔(𝑦, 𝑦).
𝑑𝑝 𝑝=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 𝑓. Assuming that 𝑓 is always
nonpositive, equivalently, that the social cost – 𝑓 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 𝑟. This dependence may be studied in a single
game or, more generally, for a specified family of games ℋ, providing that 𝑓 is meaningful
for all games in ℋ. Thus, the price of anarchy PoA𝑟 corresponding to altruism coefficient
𝑟 ≤ 1 is defined by
PoA𝑟 = sup {
𝑓(𝑦 𝑟 ) ∣ ℎ ∈ ℋ, 𝑥 is a strategy profile in ℎ,
∣
},
𝑓(𝑥) ∣ 𝑦 𝑟 is an equilibrium in ℎ𝑟
(7)
where the quotient is interpreted as 1 if the numerator and denominator are both 0 and as
∞ if only the denominator is 0. A similar definition applies to a family of population games 𝒢,
with 𝑓 replaced by a specified nonpositive social payoff 𝜙 that is meaningful for all games in
𝒢.
3 Unweighted Congestion Games
A (weighted) congestion game has a finite number 𝑁 of players, numbered from 1 to 𝑁, who
share a finite set 𝐸 of resources (represented, for example, by edges in a graph). Each player
𝑖 has a weight 𝑤𝑖 > 0 and a finite collection of strategies. Each strategy of player 𝑖 is
represented by a non-zero vector 𝑥𝑖 = (𝑥𝑖𝑒 )𝑒∈𝐸 , where each component 𝑥𝑖𝑒 is either 𝑤𝑖
(indicating that using resource 𝑒 is part of the strategy) or 0 (indicating that the strategy
does not include resource 𝑒). For a strategy profile 𝑥, the total use of resource 𝑒 is expressed
′
″
′
″
by the flow, or load, on it 𝑥𝑒 = ∑𝑁
𝑖=1 𝑥𝑖𝑒 . Two strategy profiles 𝑥 and 𝑥 such that 𝑥𝑒 = 𝑥𝑒
for all 𝑒 are said to be equivalent. The cost of using resource 𝑒 is given by 𝑐𝑒 (𝑥𝑒 ), where
𝑐𝑒 : [0, ∞) → ℝ is a resource-specific cost function. The total cost for each player 𝑖 is the sum
of the costs of the resources included in 𝑖’s strategy multiplied by the player’s weight, and
the payoff is the negative of the cost. Thus, player 𝑖’s payoff is given by
ℎ𝑖 (𝑥) = − ∑ 𝑥𝑖𝑒 𝑐𝑒 (𝑥𝑒 ).
𝑒∈𝐸
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
𝑓(𝑥) = − ∑ 𝑥𝑒 𝑐𝑒 (𝑥𝑒 ).
𝑒∈𝐸
6
(8)
With altruism coefficient 𝑟 ≤ 1, the modified payoff of player 𝑖 is given by
ℎ𝑖𝑟 (𝑥) = − ∑((1 − 𝑟)𝑥𝑖𝑒 + 𝑟𝑥𝑒 ) 𝑐𝑒 (𝑥𝑒 ).
(9)
𝑒∈𝐸
Note that the expression in parenthesis can be written as 𝑥𝑖𝑒 + 𝑟 ∑𝑗≠𝑖 𝑥𝑗𝑒 .
An unweighted congestion game is one where all players 𝑖 have unity weights, 𝑤𝑖 = 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 ℎ is always a potential game (Rosenthal 1973), with the
potential
𝑥𝑒
𝑃(𝑥) = − ∑ ∑ 𝑐𝑒 (𝑡).
(10)
𝑒∈𝐸 𝑡=1
(If 𝑥𝑒 = 0, the inner sum is 0 by convention.) As in any potential game, the set of equilibria
in ℎ coincides with the set of “local” maximum points of 𝑃, that is, strategy profiles where no
single player can increase 𝑃 by unilaterally changing his strategy. In particular, it includes the
(nonempty) set of all global maximum points of 𝑃. 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 “flow substitution” 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 𝑥 ′ and 𝑥 ″ there is a strategy profile 𝑥 that differs from 𝑥 ′
only in the strategy of a single player and satisfies min(𝑥𝑒′ , 𝑥𝑒″ ) ≤ 𝑥𝑒 ≤ max(𝑥𝑒′ , 𝑥𝑒″ ) for all 𝑒.
The monotonicity property is so called because it is equivalent to the possibility of changing
the strategies prescribed by 𝑥 ′ sequentially, one player at a time, in such a way that the flow
on each resource changes monotonically from its value at 𝑥 ′ to the value at 𝑥 ″ .
Lemma 1. The monotonicity property holds if and only if for every two strategy profiles 𝑥 ′
and 𝑥 ″ there are strategy profiles 𝑥 0 , 𝑥 1 , … , 𝑥 𝑁 such that 𝑥 0 = 𝑥 ′ , each of the other strategy
profiles 𝑥 𝑖 is either identical to its predecessor 𝑥 𝑖−1 or differs from it only in the strategy of
player 𝑖, 𝑥 𝑁 and 𝑥 ″ are equivalent, and the sequence of flows 𝑥𝑒0 , 𝑥𝑒1 , … , 𝑥𝑒𝑁 is monotone for
all 𝑒.
Proof. The sufficiency of the condition (“if”) is proved by setting 𝑥 = 𝑥 𝑖 , where 𝑖 is the
smallest index for which 𝑥 𝑖 ≠ 𝑥 ′ (which must exist if 𝑥 ′ and 𝑥 ″ are non-equivalent). To prove
necessity (“only if”), construct for given strategy profile 𝑥 ′ and 𝑥 ″ a finite sequence 𝑥̅ 0 , 𝑥̅ 1 , …
as follows: 𝑥̅ 0 = 𝑥 ′ , and for each 𝑘 ≥ 1, 𝑥̅ 𝑘 is a strategy profile that satisfies the conditions
specified by Definition 1 for 𝑥 except that 𝑥 ′ there is replaced with 𝑥̅ 𝑘−1 . Note that
𝑥̅𝑒𝑘 ≠ 𝑥̅𝑒𝑘−1 for at least one resource 𝑒 and that, for all 𝑒, 𝑥̅𝑒0 ≤ 𝑥̅𝑒1 ≤ ⋯ ≤ 𝑥𝑒″ or 𝑥̅𝑒0 ≥ 𝑥̅𝑒1 ≥
⋯ ≥ 𝑥𝑒″ . These facts imply that the sequence cannot be extended indefinitely, because for
7
some 𝐾 ≥ 0 the strategy profile 𝑥̅ 𝐾 and 𝑥 ″ must be equivalent. Now, for 0 ≤ 𝑖 ≤ 𝑁, define
𝑥 𝑖 as the strategy profile that prescribes the same strategies as 𝑥̅ 𝐾 to all players with index 𝑖
or lower, and otherwise agrees with 𝑥 ′ . It is not difficult to see that, because (𝑥𝑒′ =)𝑥̅𝑒0 , 𝑥̅𝑒1 ,
… , 𝑥̅𝑒𝐾 (= 𝑥𝑒″ ) is monotone for each 𝑒, the same is true for (𝑥𝑒′ =)𝑥𝑒0 , 𝑥𝑒1 , … , 𝑥𝑒𝑁 (= 𝑥𝑒″ ).
∎
Proposition 3. If an unweighted congestion game with nondecreasing cost functions has the
monotonicity property, then every equilibrium maximizes the potential 𝑃.
Proof. Consider a game as above that has the monotonicity property. Let 𝑥 ′ be an
equilibrium, 𝑥 ″ any other strategy profile, and 𝑥 0 , 𝑥 1 , … , 𝑥 𝑁 strategy profiles as in Lemma 1.
Since 𝑥 𝑁 and 𝑥 ″ are equivalent, and therefore 𝑃(𝑥 𝑁 ) = 𝑃(𝑥 ″ ), to prove that 𝑃(𝑥 ′ ) ≥ 𝑃(𝑥 ″ )
it suffices to show that 𝑃(𝑥 𝑖 ) − 𝑃(𝑥 𝑖−1 ) is nonpositive for all 1 ≤ 𝑖 ≤ 𝑁.
For each 1 ≤ 𝑖 ≤ 𝑁, the strategy profile 𝑥 𝑖 can differ from 𝑥 𝑖−1 only in the strategy of player
𝑖. Therefore, the difference 𝑃(𝑥 𝑖 ) − 𝑃(𝑥 𝑖−1 ) is equals to the difference between the
player’s payoff in 𝑥 𝑖 and 𝑥 𝑖−1 , which is
(11)
𝑖
𝑖−1
− ∑ (𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒𝑖 ) − 𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒𝑖−1 )).
𝑒∈𝐸
𝑖
𝑖−1
𝑖
𝑖−1
In this sum, each term with 𝑥𝑖𝑒
= 𝑥𝑖𝑒
is clearly zero. If 𝑥𝑖𝑒
= 1 and 𝑥𝑖𝑒
= 0, then it
′
0
1
𝑖
′
𝑖−1
𝑖
follows from the monotonicity of (𝑥𝑒 =)𝑥𝑒 , 𝑥𝑒 , … , 𝑥𝑒 that 𝑥𝑒 ≤ 𝑥𝑒 = 𝑥𝑒 − 1. Similarly, if
𝑖
𝑖−1
𝑥𝑖𝑒
= 0 and 𝑥𝑖𝑒
= 1, then 𝑥𝑒′ ≥ 𝑥𝑒𝑖−1 = 𝑥𝑒𝑖 + 1. Since 𝑐𝑒 is nondecreasing, in all three cases
𝑖
𝑖−1
𝑖
𝑖
𝑖−1 )
𝑖−1
− (𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒𝑖 ) − 𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒𝑖−1 )) ≤ − (𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒′ + 𝑥𝑖𝑒
− 𝑥𝑖𝑒
− 𝑥𝑖𝑒
𝑐𝑒 (𝑥𝑒′ )).
The sum over 𝑒 of the expression on the right is therefore greater than or equal to (11). This
sum gives the change in player 𝑖’s payoff that would result if he deviated directly from 𝑥 ′ , by
moving from 𝑥𝑖′ to 𝑥𝑖𝑖 . Since 𝑥 is an equilibrium, this change is nonpositive, with implies the
same for (11).
∎
For an unweighted congestion game ℎ, consider the social payoff defined as the aggregate
payoff 𝑓 (and the social cost defined as the aggregate cost −𝑓). It follows from (6), (8) and
(10) that, for altruism coefficient 𝑟 ≤ 1, the modified game ℎ𝑟 has the potential
𝑥𝑒
𝑃
𝑟 (𝑥)
𝑥𝑒
= − ∑ ((1 − 𝑟) ∑ 𝑐𝑒 (𝑡) + 𝑟 𝑥𝑒 𝑐𝑒 (𝑥𝑒 )) = − ∑ ∑ 𝑐̂𝑒𝑟 (𝑡),
𝑒∈𝐸
𝑡=1
(12)
𝑒∈𝐸 𝑡=1
where the function 𝑐̂𝑒𝑟 is defined by
𝑐̂𝑒𝑟 = (1 − 𝑟) 𝑐𝑒 + 𝑟 𝑀𝐶𝑒
and the function 𝑀𝐶𝑒 , the (discrete version of the) marginal social cost, is defined by
𝑀𝐶𝑒 (𝑡) = 𝑡 𝑐𝑒 (𝑡) − (𝑡 − 1) 𝑐𝑒 (𝑡 − 1),
𝑡 = 1,2, ….
The game ℎ𝑟 is generally not a congestion game. However, comparison of (10) with (12)
shows that 𝑃𝑟 is also the potential of some such game, namely, the unweighted congestion
8
game ℎ̂𝑟 that differs from (the original, unmodified one) ℎ only in that the cost function 𝑐𝑒 of
each resource 𝑒 is replaced with 𝑐̂𝑒𝑟 . This observation leads to the following comparativestatics result.
Theorem 1. Let ℎ be an unweighted congestion game where the cost functions and marginal
social costs are nondecreasing. For the aggregate payoff 𝑓 as the social payoff, and altruism
coefficients 𝑟 and 𝑠 with 0 ≤ 𝑟 < 𝑠 ≤ 1, let 𝑦 𝑟 and 𝑦 𝑠 be equilibria in the modified games
ℎ𝑟 and ℎ 𝑠 , respectively. If ℎ has the monotonicity property, then
𝑓(𝑦 𝑟 ) ≤ 𝑓(𝑦 𝑠 ).
(13)
If, in addition, 𝑠 = 1, then 𝑓(𝑦 𝑠 ) if the maximum of the social payoff function.
Proof. As indicated, the modified games ℎ𝑟 and ℎ 𝑠 share their potentials, 𝑃𝑟 and 𝑃 𝑠 , and
therefore also their sets of equilibria, with the unweighted congestion games ℎ̂𝑟 and ℎ̂ 𝑠 . The
latter differ from ℎ only in the cost functions, and therefore have the monotonicity property
if and only if ℎ has it. In this case, it follows from Proposition 3 that the equilibria 𝑦 𝑟 and 𝑦 𝑠
maximize 𝑃𝑟 and 𝑃 𝑠 , respectively, because the assumption concerning the costs and
marginal social costs in ℎ implies that 𝑐̂𝑒𝑟 and 𝑐̂𝑒𝑠 are nondecreasing for every resource 𝑒.
Inequality (13) now follows from Proposition 1. If 𝑠 = 1, then 𝑃 𝑠 = 𝑓, so that 𝑦 𝑠 is a
maximum point of the social payoff.
∎
For a game ℎ with nonnegative costs, Theorem 1 implies that, if the monotonicity property
holds, then the price of anarchy (see Section ‎2.2) 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 ℋ 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
PoA𝑟 is nonincreasing for 0 ≤ 𝑟 ≤ 1, with 1 ≤ PoA𝑟 ≤ 1/𝑟.
Proof. Consider any 𝑟 and 𝑠 with 0 ≤ 𝑟 ≤ 𝑠 ≤ 1, game ℎ ∈ ℋ, strategy profile 𝑥 in ℎ, and
equilibria 𝑦 𝑟 and 𝑦 𝑠 in ℎ𝑟 and ℎ 𝑠 , respectively. By the nonnegativity and monotonicity of the
cost functions, 𝑓 ≤ 𝑃 ≤ 0, and therefore 𝑓 ≤ 𝑃𝑟 ≤ 𝑟𝑓. Since 𝑦 𝑟 maximizes 𝑃𝑟 (see the
proof of Theorem 1), these inequalities give
𝑓(𝑥) ≤ 𝑃𝑟 (𝑥) ≤ 𝑃𝑟 (𝑦 𝑟 ) ≤ 𝑟𝑓(𝑦 𝑟 ).
Therefore, by (13), 𝑓(𝑦 𝑠 )/𝑓(𝑥) ≤ 𝑓(𝑦 𝑟 )/𝑓(𝑥) ≤ 1/𝑟. From theses inequalities the result
easily follows. (By definition of PoA𝑟 , the first inequality gives 𝑓(𝑦 𝑠 )/𝑓(𝑥) ≤ PoA𝑟 .)
∎
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 ℎ has two players, four resources and
four strategies, which are: resource 𝑒1 alone, and any two of resources 𝑒2 , 𝑒3 and 𝑒4 . The
9
cost functions are: 𝑐𝑒1 (𝑡) = 𝑡 + 9, 𝑐𝑒2 (𝑡) = 𝑡 + 2.5 and 𝑐𝑒3 (𝑡) = 𝑐𝑒4 (𝑡) = 2𝑡 + 1. A strategy
profile minimizes the aggregate cost if and only if one player uses 𝑒2 and 𝑒3 and the other
player uses 𝑒2 and 𝑒4 . Such a strategy profile is also an equilibrium in the modified game, for
all 0 ≤ 𝑟 ≤ 1. For 0.75 ≤ 𝑟 ≤ 1, there is a second kind of equilibrium, where one player
uses 𝑒1 and the other player uses 𝑒3 and 𝑒4 . Therefore, the price of anarchy satisfies
PoA𝑟 = 1 for 0 ≤ 𝑟 < 0.75 and is greater than 1 (specifically, = 16/15) for 0.75 ≤ 𝑟 ≤ 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 ℋ 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
𝑐𝑒 (𝑡) = 𝑡 for all resources 𝑒), 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 ≤ 𝑟 ≤ 1,
and is given by
PoA𝑟 =
5 + 4𝑟
.
2+𝑟
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 PoA𝑟 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 𝑜 and 𝑑, whose directed edges represent the resources. It is assumed that none of
the edges and vertices in the network is “redundant”, that is, each of them belongs to at
least one route from 𝑜 to 𝑑. 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 𝐺, where each vertex and each edge belongs to at least one route, and
(ii) an assignment of an allowable direction to each edge 𝑒 in 𝐺, which must be that in which
some route in 𝐺 traverses 𝑒 (Milchtaich 2013).4 The set of strategies consists of all allowable
routes in 𝐺, 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 𝐺 that differ in the number of players, costs functions or edge directions
are grouped together. Such a classification may be useful for identifying certain
“topological” 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 𝐺 has the monotonicity property if all
symmetric unweighted network congestion games on 𝐺 have the monotonicity property.
Lemma 2. If an undirected two-terminal network 𝐺 is obtained by connecting two or more
networks in series, then 𝐺 has the monotonicity property if and only if each of the
constituent networks has the property.
Proof. Assigning allowable directions to the edges in 𝐺 is equivalent to assigning directions in
each of the constituent networks 𝐺̅ , and a route in 𝐺 is allowable if and only if the section
that passes through each 𝐺̅ is allowable there. Therefore, 𝐺 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 𝑥 ′ and 𝑥 ″ , there is a strategy profile 𝑥 that differs
from 𝑥 ′ only in the strategy of a single player, who only changes his route in one constituent
network 𝐺̅ , such that min(𝑥𝑒′ , 𝑥𝑒″ ) ≤ 𝑥𝑒 ≤ max(𝑥𝑒′ , 𝑥𝑒″ ) for all 𝑒 in 𝐺̅ . Clearly, this condition
holds if and only if every 𝐺̅ have the monotonicity property.
∎
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 𝑟 = 1, a strategy profile is an equilibrium in this
game if and only if it is “locally” 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 “global” 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 𝑁 = 2, a strategy profile 𝑥 as in Definition 1 does not exist for a strategy
profile 𝑥 ′ where one player uses the route 𝑒1 𝑒4 and the other uses 𝑒2 𝑒5 and a strategy
profile 𝑥 ″ where the routes are 𝑒1 𝑒5 and 𝑒3 . Moreover, such 𝑥 ′ and 𝑥 ″ 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 “local” minimum points of the aggregate cost. However, the aggregate
cost at 𝑥′ is higher than at 𝑥 ″ , 8 instead of 7. An almost identical pair of equilibria exists in
11
𝑜
𝑜
𝑒1 𝑒2
𝑒2
𝑒1
𝑜
𝑒1
𝑜
𝑒2
𝑒1
𝑒3
𝑒3
𝑣
𝑒4
𝑒5
𝑢
𝑣
𝑒6
𝑒3
𝑢
𝑒4 𝑒5
𝑒5
𝑒2
𝑣 𝑢
𝑒4
𝑒6
𝑒5
𝑒4
𝑒3
𝑑
𝑑
𝑑
𝑑
(a)
(b)
(c)
(d)
Figure 1. Symmetric unweighted network congestion games. 𝑵 unity-weight players choose routes from 𝒐 to
𝒅. The cost functions are: (a) 𝒄𝒆𝟏 (𝒕) = 𝒄𝒆𝟓 (𝒕) = 𝒕𝟐 , 𝒄𝒆𝟐 (𝒕) = 𝒄𝒆𝟒 (𝒕) = 𝟑𝒕 and 𝒄𝒆𝟑 (𝒕) = 𝟓𝒕, (b) the same, and
𝒄𝒆𝟔 (𝒕) = 𝟎. 𝟓𝒕, (c) 𝒄𝒆 (𝒕) = 𝒕 for all edges 𝒆, 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 𝑥 ″ , the first route is 𝑒1 𝑒6 𝑒5 . Similarly, in Figure 1c (again, a non-series-parallel network),
only the equilibria where the two players’ routes are 𝑒1 𝑒5 and 𝑒2 𝑒6 are socially optimal.
With 𝑒1 𝑒3 𝑒6 and 𝑒2 𝑒4 𝑒5 , 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 𝑁
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, 𝑜 or 𝑑, 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 𝐺 has the augmented monotonicity property (AMP) if the augmented
network 𝐺 ∗ obtained by connecting 𝐺 in parallel with a single edge 𝑒 ∗ has the monotonicity
property. The edge 𝑒 ∗ may be interpreted as representing a common outside option, an
alternative to using the network 𝐺.
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 𝐺 is obtained by
connecting in series an extension-parallel network 𝐺̅ that has the AMP and a single edge 𝑒,
then 𝐺 also has the AMP. This is because the set of routes (equivalently, allowable routes;
see Corollary A2 in the Appendix, condition (ii)) in 𝐺 is obtained from that in 𝐺̅ simply by the
addition of 𝑒 (and its non-terminal end vertex) to each route. Therefore, the flow on 𝑒 and
the flow on 𝑒 ∗ sum up to the number of players 𝑁 for every strategy profile in the
augmented network 𝐺̅∗ , 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 𝐺 obtained by connecting in parallel two extension-parallel networks 𝐺̅ and 𝐺̃ that
have the AMP, and show that the augmented network 𝐺 ∗ (consisting of 𝐺̅ , 𝐺̃ and a single
edge 𝑒 ∗ all connected in parallel) has the monotonicity property.
For any number of players 𝑁, consider two non-equivalent strategy profiles 𝑥′ and 𝑥 ″ where
the players use routes in 𝐺 ∗ . Let 𝑥̅ ′ and 𝑥̅ ″ be the strategy profiles obtained from 𝑥′ and 𝑥 ″
by replacing each route that lies outside 𝐺̅ ∗ (that is, in 𝐺̃ ) with the single-edge route 𝑒 ∗ . In
particular, for 𝑥′, 𝑥̅𝑒′ ∗ gives the number of players who are not using a route in 𝐺̅ , and
similarly for 𝑥 ″ and 𝑥̅𝑒″∗ . Since 𝐺̅ has the AMP, if 𝑥̅ ′ and 𝑥̅ ″ are non-equivalent, then applying
Definition 1 to them (instead of 𝑥′ and 𝑥 ″ ) gives a strategy profile 𝑥̅ , which only involves
routes in 𝐺̅ ∗ . For 𝐺̃ , the strategy profiles 𝑥̃ ′ , 𝑥̃ ″ and (if the last two are non-equivalent) 𝑥̃ are
constructed analogously. Using these ingredients, a strategy profile 𝑥 satisfying the
conditions in Definition 1 is constructed below.
Note, first, that the strategy profiles 𝑥̅ ′ and 𝑥̅ ″ are non-equivalent or 𝑥̃ ′ and 𝑥̃ ″ are nonequivalent. Otherwise, both the flow on each edge in 𝐺̅ and the total number of players
using routes in that network (which is 𝑁 minus the number of players not doing so) would
be the same for 𝑥′ and 𝑥 ″ , and similarly for 𝐺̃ . However, the conclusion implies that the
number of players using 𝑒 ∗ is also the same for 𝑥′ and 𝑥 ″ , which contradicts their assumed
non-equivalence. By symmetry, it suffices to consider the case where 𝑥̅ ′ and 𝑥̅ ″ are nonequivalent, so that a strategy profile 𝑥̅ as above and a single player 𝑖 with 𝑥̅𝑖 ≠ 𝑥̅𝑖′ exist.
Suppose, first, that 𝑥̅𝑖′ is a route in 𝐺̅ (hence, 𝑥̅𝑖′ = 𝑥𝑖′ ). If 𝑥̅𝑖 is also a route there, then define
𝑥 by 𝑥𝑖 = 𝑥̅𝑖 . (The other strategies in 𝑥 are as in 𝑥 ′ .) If 𝑥̅𝑖 is 𝑒 ∗ , then 𝑥̅𝑒″∗ > 𝑥̅𝑒′ ∗ by definition
of the monotonicity property. If in addition 𝑥̃𝑒″∗ ≥ 𝑥̃𝑒′ ∗ , then necessarily 𝑥𝑒″∗ > 𝑥𝑒′ ∗ , and in this
case, set 𝑥𝑖 to be 𝑒 ∗ . If 𝑥̃𝑒″∗ < 𝑥̃𝑒′ ∗ , then 𝑥̃ ′ and 𝑥̃ ″ are not equivalent and there is a player 𝑗
with 𝑥̃𝑗 ≠ 𝑥̃𝑗′ where 𝑥̃𝑗 is not 𝑒 ∗ (but rather a route in 𝐺̃ ). In this case, define 𝑥 by 𝑥𝑖 = 𝑥̃𝑗 or
𝑥𝑗 = 𝑥̃𝑗 if 𝑥̃𝑗′ is or is not 𝑒 ∗ , respectively. Thus, in the first case, player 𝑖 moves to the route
that player 𝑗 (who may or may not be 𝑖) was supposed to move to.
In remains to consider the case where 𝑥̅𝑖′ is 𝑒 ∗ and 𝑥̅𝑖 is a route in 𝐺̅ , which by definition of
the monotonicity property implies that 𝑥̅𝑒″∗ < 𝑥̅𝑒′ ∗ . If in addition 𝑥̃𝑒″∗ ≤ 𝑥̃𝑒′ ∗ , then necessarily
𝑥𝑒″∗ < 𝑥𝑒′ ∗ , and in this case, choose any player 𝑗 for whom 𝑥𝑗′ is 𝑒 ∗ and define 𝑥 by 𝑥𝑗 = 𝑥̅𝑖 . If
𝑥̃𝑒″∗ > 𝑥̃𝑒′ ∗ , then 𝑥̃ ′ and 𝑥̃ ″ are not equivalent and there is a player 𝑗 with 𝑥̃𝑗 ≠ 𝑥̃𝑗′ where 𝑥̃𝑗′ is
13
a route in 𝐺̃ . In this case, set 𝑥𝑗 = 𝑥̅𝑖 or 𝑥𝑗 = 𝑥̃𝑗 if 𝑥̃𝑗′ is or is not 𝑒 ∗ , 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 𝐺 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 𝑒3 , the “expanded” figure-eight network obtained from the previous
network by replacing the vertex 𝑣 with two vertices connected by an edge, and the
Wheatstone network, which is the undirected version of the network in Figure 1b minus 𝑒3
(Milchtaich 2006b, Proposition 4). Therefore, connecting 𝐺 in parallel with a single edge 𝑒 ∗
results in an undirected network 𝐺 ∗ for which a two-player game as in Example 2, that is, a
game without the monotonicity property, exists.
∎
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 𝑒3 in Figure 1c with (ii) the
network resulting from deleting edge 𝑒3 in Figure 1d. Network (ii) is extension-parallel, and
therefore has the monotonicity property. The addition of 𝑒6 (and 𝑣) to each route in this
network that includes either 𝑒2 or 𝑒4 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 𝑒6 must always be equal
to the number of players 𝑁 minus the flow on 𝑒5 , 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 𝐺 is that, in every symmetric unweighted network congestion game on 𝐺, 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 players’ 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 𝐺 efficient if, in every symmetric unweighted
network congestion game on 𝐺, every equilibrium minimizes the makespan, that is, it is
optimal with respect to the maximum (rather than the sum) of the players’ 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 predecessors’ 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 𝐺 if and only
if 𝐺 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 ‎3 (hereafter, a game with unsplittable flow) by replacing the strategy set of each
player 𝑖 with its convex hull. Thus, each strategy 𝑥𝑖 = (𝑥𝑖𝑒 )𝑒∈𝐸 is a convex combination of
strategies as in Section ‎3 (hereafter, “pure” strategies), with coefficients that express the
fraction of the player’s weight 𝑤𝑖 “shipped” on each of them. The resulting amount of use by
player 𝑖 of each resource 𝑒 is given by the corresponding component 0 ≤ 𝑥𝑖𝑒 ≤ 𝑤𝑖 .
An interesting special case is that of linear congestion games with splittable flow, where
there are constants 𝑎𝑒 > 0 and 𝑏𝑒 for each resource 𝑒 such that
𝑐𝑒 (𝑡) = 𝑎𝑒 𝑡 + 𝑏𝑒 .
The linearity of the cost functions implies that the game is a potential game, as it is not
difficult to check that the function 𝑃 defined as follows (Fotakis et al. 2005) is an (exact)
potential:
𝑃(𝑥) = − ∑ (𝑎𝑒
𝑒∈𝐸
2
𝑥𝑒2 + ∑𝑁
𝑖=1 𝑥𝑖𝑒
+ 𝑏𝑒 𝑥𝑒 ).
2
Observe that 𝑃 ≥ 𝑓, where 𝑓 is the aggregate payoff (8). With the latter as the social payoff,
and altruism coefficient 𝑟 ≤ 1, the modified game has the following potential (see (6)):
𝑃𝑟 (𝑥) = − ∑ (𝑎𝑒
𝑒∈𝐸
2
(1 + 𝑟)𝑥𝑒2 + (1 − 𝑟) ∑𝑁
𝑖=1 𝑥𝑖𝑒
+ 𝑏𝑒 𝑥𝑒 ).
2
Clearly, there exists at least one strategy profile maximizing 𝑃𝑟 , 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 𝑟 ≠ 1, examines the dependence of the corresponding social
payoff on the altruism coefficient.
Theorem 2. For a linear congestion game with splittable flow ℎ, and the aggregate payoff 𝑓
as the social payoff, the modified game ℎ𝑟 has a unique equilibrium 𝑦 𝑟 for every −1 ≤ 𝑟 <
1, and the function 𝑟 ↦ 𝑦 𝑟 is continuous. For ℎ1 , uniqueness does not necessarily hold but
every equilibrium 𝑦1 maximizes 𝑓. The function 𝜋: [−1,1] → ℝ (well) defined by
𝜋(𝑟) = 𝑓(𝑦 𝑟 )
is absolutely continuous and there is a partition of [−1,1] into finitely many intervals such
that 𝜋 is either constant or strictly increasing in each interval.
Proof. To prove that for −1 ≤ 𝑟 < 1 there is only one equilibrium in ℎ𝑟 , and therefore only
one maximum point of the potential 𝑃𝑟 , it suffices to show that every equilibrium 𝑦 is a
strict global maximum point of 𝑃𝑟 . Thus, it has to be shown that for every strategy profile
𝑥 ≠ 𝑦 the function Λ: [0,1] → ℝ defined by Λ(λ) = 𝑃𝑟 (λ𝑥 + (1 − 𝜆)𝑦) satisfies Λ(0) >
Λ(λ) for all 0 < λ ≤ 1. For this, it suffices that Λ′ (0) ≤ 0 and Λ″ < 0. Substitution in (14)
and derivation give:
𝑁
′ (0)
Λ
= − ∑ ∑(𝑎𝑒 (1 + 𝑟)𝑦𝑒 + 𝑎𝑒 (1 − 𝑟)𝑦𝑖𝑒 + 𝑏𝑒 )(𝑥𝑖𝑒 − 𝑦𝑖𝑒 ).
(15)
𝑖=1 𝑒∈𝐸
The expression on the right-hand side is equal to the sum
𝑁
∑
𝑖=1
𝑑
|
ℎ𝑟 ( 𝑦 ∣ λ𝑥𝑖 + (1 − 𝜆)𝑦𝑖 ),
𝑑λ λ=0+ 𝑖
whose terms are all nonpositive because 𝑦 is an equilibrium in ℎ𝑟 . This proves that
Λ′ (0) ≤ 0. The second inequality above holds even without the equilibrium assumption
(which shows that the potential 𝑃𝑟 is a strictly concave function on the set of strategy
profiles):
𝑁
″
Λ = −(1 − 𝑟) ∑ 𝑎𝑒 ∑(𝑥𝑖𝑒 − 𝑦𝑖𝑒 )2 − (1 + 𝑟) ∑ 𝑎𝑒 (𝑥𝑒 − 𝑦𝑒 )2 < 0.
𝑒∈𝐸
𝑖=1
𝑒∈𝐸
For 𝑟 = 1, Λ″ ≤ 0 (and equality holds if and only if the flows on all resources are the same
for 𝑥 and 𝑦). Therefore, every equilibrium in ℎ1 is a maximum point of the potential 𝑃1
(which coincides with 𝑓). It follows that the projection on the first two coordinates of the set
𝒜 = { (𝑟, 𝛼, 𝑦) ∣ −1 ≤ 𝑟 ≤ 1, 𝛼 = 𝑓(𝑦), 𝑦 is an equilibrium in ℎ𝑟 }
is the graph of a function, denoted 𝜋: [−1,1] → ℝ. By the Tarski–Seidenberg theorem, a
sufficient condition for 𝜋 to be continuous and semialgebraic is that the set 𝒜 is closed
(equivalently, compact) and semialgebraic. This condition holds because, as shown, 𝑦 is an
equilibrium in ℎ𝑟 if and only if it maximizes 𝑃𝑟 , hence if and only if the (polynomial)
expression in (15) is nonpositive for all profiles 𝑥 of pure strategies (which, by linearity,
16
𝑜1
𝑑3
𝑑2
𝑜3
𝑑1
𝑜2
Figure 2. A three-player congestion game with splittable flow. Each player 𝒊 has to ship a unit flow from 𝒐𝒊 to
𝒅𝒊 , 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 𝟐.
automatically implies the same for mixed strategies). The compactness of 𝒜 also implies
that, on the half-open interval [−1,1), the function mapping each 𝑟 to the corresponding
unique equilibrium 𝑦 𝑟 is continuous.
Since 𝜋 is a continuous semialgebraic function, for every −1 ≤ 𝑟 ≤ 1 there is some 𝜖 > 0, a
pair of positive integers 𝑝𝑅 and 𝑝𝐿 and a pair of analytic functions 𝜉𝑅 , 𝜉𝐿 : (−𝜖, 𝜖) → ℝ such
that 𝜋(𝑠) = 𝜉𝑅 ((𝑠 − 𝑟)1/𝑝𝑅 ) for all 𝑟 ≤ 𝑠 ≤ min{1, 𝑟 + 𝜖 𝑝𝑅 /2} and 𝜋(𝑠) = 𝜉𝐿 ((𝑟 − 𝑠)1/𝑝𝐿 )
for all max{−1, 𝑟 − 𝜖 𝑝𝐿 /2} ≤ 𝑠 ≤ 𝑟 (see Bochnak et al. 1998, Proposition 8.1.12). By
decreasing 𝜖, if necessarily, it can be guaranteed that each of the analytic functions 𝜉𝑅 and
𝜉𝐿 is either constant or strictly monotone. The collection of all intervals of the form
(𝑟 − 𝜖 𝑝𝐿 /2, 𝑟 + 𝜖 𝑝𝑅 /2), where 𝑟 varies over all points in [−1,1] and 𝜖, 𝑝𝑅 and 𝑝𝐿 vary
correspondingly, constitutes an open cover of [−1,1]. Any finite subcover yields a finite set
of points −1 = 𝑟0 < 𝑟1 < ⋯ < 𝑟𝐿 = 1 such that, for all 1 ≤ 𝑙 ≤ 𝐿, the restriction of 𝜋 to the
interval [𝑟𝑙−1 , 𝑟𝑙 ] is continuous and either constant or strictly monotone and has a
continuous derivative in (𝑟𝑙−1 , 𝑟𝑙 ). These properties imply that 𝜋 is absolutely continuous in
the interval. It remains to observe that, by Proposition 1, it cannot be strictly decreasing
there.
∎
The next example illustrates the theorem.
Example 3. In a three-player linear congestion game with splittable flow, 𝐸 is the set of all
edges in the undirected graph shown in Figure 2. The weight of each player 𝑖 is 1, and the
player’s “pure” strategies are the (simple) paths in the graph that connect 𝑖’s origin vertex 𝑜𝑖
with the destination vertex 𝑑𝑖 .7 The cost of each edge 𝑒 is equal to the flow on it 𝑥𝑒 , except
7
Note that this (asymmetric) game differs from the network congestion game considered in
Section ‎3.1 in that edges can be traversed in both directions.
17
for the three “thick” edges, for which the cost is 𝑥𝑒 + 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 ≤ 𝑟 ≤ 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, −1 ≤ 𝑟 < 1/3, it is not an equilibrium.
Instead, the unique equilibrium is for every player to ship only (9 − 𝑟)/(10 − 4𝑟) of the
weight on the short path, and the rest on the longer, four-edge path. The corresponding
social payoff 𝜋(𝑟), which is the negative of the aggregate cost, is given by
𝜋(𝑟) = − (169 (
1−𝑟 2
) + 8).
10 − 4𝑟
As 𝑟 increases from −1 to 1/3, 𝜋(𝑟) 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, 𝑏𝑒 ≥ 0 for every resource 𝑒), 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 𝑟 = 0), the price of anarchy for
games as above does not exceed 3/2 (Cominetti et al. 2009).
Corollary 2. For the family ℋ of all linear congestion games with splittable flow and
nonnegative cost functions, as well as every subfamily or individual game, the price of
anarchy PoA𝑟 is nonincreasing for −1 ≤ 𝑟 ≤ 1, with
1 ≤ PoA𝑟 ≤
1
1
+ .
1+𝑟 2
(16)
Proof. Consider, more generally, any 𝑁-player congestion game with splittable flow ℎ where
every cost function 𝑐𝑒 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 −𝑓 satisfies
(1 − max 𝛣𝑁 (𝑐𝑒 ))(−𝑓(𝑦)) ≤ −𝑓(𝑥)
𝑒
(17)
for any equilibrium 𝑦, any strategy profile 𝑥 and any assignment of a number 𝛣𝑁 (𝑐𝑒 ) to each
cost function 𝑐𝑒 such that
𝑠 (𝑐𝑒 (𝑡) − 𝑐𝑒 (𝑠)) + (𝑠 2 /4 − (𝑡 − 𝑠/2)2 /𝑁) 𝑐𝑒′ (𝑡)
.
𝑡𝑐𝑒 (𝑡)
𝑠,𝑡>0
𝛣𝑁 (𝑐𝑒 ) ≥ sup
(18)
A straightforward generalization of the proofs of these results yields a similar conclusion in
the more general case where 𝑦 is an equilibrium in the modified game ℎ𝑟 , for any
−1 ≤ 𝑟 ≤ 1. The only difference is that the numerator in inequality (18) is replaced with
𝑠 (𝑐𝑒 (𝑡) − 𝑐𝑒 (𝑠)) + (𝑟(𝑠 − 𝑡)𝑡 + (1 − 𝑟)(𝑠 2 /4 − (𝑡 − 𝑠/2)2 /𝑁)) 𝑐𝑒′ (𝑡).
18
In the linear case, where the derivative 𝑐𝑒′ is constant8 and therefore 𝑐𝑒 (𝑡) ≥ 𝑡𝑐𝑒′ , the
modified inequality holds for
(𝑠 (𝑡 − 𝑠) + 𝑟(𝑠 − 𝑡)𝑡 + (1 − 𝑟)(𝑠 2 /4 − (𝑡 − 𝑠/2)2 /𝑁)) 𝑐𝑒′
𝑡 2 𝑐𝑒′
𝑠,𝑡>0
1−𝑟
= sup(𝛾 (1 − 𝛾) + 𝑟(𝛾 − 1) + (1 − 𝑟)(𝛾 2 /4 − (1 − 𝛾/2)2 /𝑁)) =
.
3 + 𝑟 + 4/(𝑁 − 1)
𝛾>0
𝛣𝑁 (𝑐𝑒 ) = sup
Substitution in (17) and rearrangement give
1
1
−𝑓(𝑦) ≤ (
+ ) (−𝑓(𝑥)).
1 + 𝑟 + (1 − 𝑟)/𝑁 2
∎
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 ‎3, PoA𝑟 is strictly
increasing for 0 ≤ 𝑟 ≤ 1.
5 Nonatomic Congestion Games
A nonatomic congestion game has an infinite set, or population, of players 𝐼 (e.g., the unit
interval [0,1]), endowed with a nonatomic probability measure 𝜇, the population measure
(e.g., Lebesgue measure). The players share a finite set 𝐸 of resources and a (finite)
collection of (“pure”) strategies, which are non-zero binary vectors of the form 𝜎 = (𝜎𝑒 )𝑒∈𝐸 .
A strategy profile is any assignment of a strategy 𝜎(𝑖) = (𝜎𝑒 (𝑖))𝑒∈𝐸 to each player 𝑖 such
that, for each resource 𝑒, the set of all players 𝑖 with 𝜎𝑒 (𝑖) = 1 is measurable. The flow on
resource 𝑒, 𝑦𝑒 , is the measure of this set, and is therefore given by
𝑦𝑒 = ∫ 𝜎𝑒 (𝑖) 𝑑𝜇(𝑖).
The vector 𝑦 = (𝑦𝑒 )𝑒∈𝐸 is the population strategy. It lies in the convex hull of the set of
strategies, denoted 𝑋, which may be thought of as the collection of all mixed strategies. The
cost of using each resource 𝑒 is given by 𝑐𝑒 (𝑦𝑒 ), where 𝑐𝑒 : [0, ∞) → ℝ is a strictly increasing
and continuously differentiable cost function. The cost of using a strategy 𝜎 = (𝜎𝑒 )𝑒∈𝐸 is the
sum of the costs of all resources 𝑒 with 𝜎𝑒 = 1, and the associated payoff is the negative of
the cost. A natural extension to mixed strategies assigns to each 𝑥 = (𝑥𝑒 )𝑒∈𝐸 ∈ 𝑋 the payoff
𝑔(𝑥, 𝑦) = − ∑ 𝑥𝑒 𝑐𝑒 (𝑦𝑒 ).
(19)
𝑒∈𝐸
The mean payoff 𝜙(𝑦) is defined by
𝜙(𝑦) = 𝑔(𝑦, 𝑦) = − ∑ 𝑦𝑒 𝑐𝑒 (𝑦𝑒 ).
(20)
𝑒∈𝐸
8
For linear congestion games with splittable flow as defined in this paper, the derivative is a positive
constant, 𝑎𝑒 > 0. However, the bound (16) and its proof here apply also if 𝑎𝑒 = 0 is allowed for.
19
Eq. (19) defines a population game 𝑔 (see Section ‎2), where the strategy space is 𝑋. The
mean payoff 𝜙 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 𝑋̂, the cone of 𝑋. With
altruism coefficient 0 ≤ 𝑟 ≤ 1, the modified game 𝑔𝑟 (which is defined by (2)) is given by
𝑔𝑟 (𝑥, 𝑦) = − ∑ 𝑥𝑒 𝑐̂𝑒𝑟 (𝑦𝑒 ) = − ∑ 𝑥𝑒 (𝑐𝑒 (𝑦𝑒 ) + 𝑟 𝑦𝑒 𝑐𝑒′ (𝑦𝑒 )),
𝑒∈𝐸
(21)
𝑒∈𝐸
where
𝑐̂𝑒𝑟 = (1 − 𝑟) 𝑐𝑒 + 𝑟 𝑀𝐶𝑒
and 𝑀𝐶𝑒 : [0, ∞) → ℝ is the marginal social cost defined by
𝑀𝐶𝑒 (𝑡) =
𝑑
𝑡𝑐 (𝑡) = 𝑐𝑒 (𝑡) + 𝑡 𝑐𝑒′ (𝑡).
𝑑𝑡 𝑒
(22)
If 𝑀𝐶𝑒 is strictly increasing and continuously differentiable for every resource 𝑒, then 𝑔𝑟
may also be viewed as the population game representing a nonatomic congestion game,
which differs from the original one in that each cost function 𝑐𝑒 is replaced with 𝑐̂𝑒𝑟 . A
sufficient condition for 𝑀𝐶𝑒 to be strictly increasing in any interval is that 𝑐𝑒 is convex there.
The population game 𝑔 defined above is a potential game, as it is easy to see that the (well
known; see Milchtaich 2004, Section 5) function 𝛷: 𝑋̂ → ℝ defined by
𝑥̂𝑒
𝛷(𝑥̂) = − ∑ ∫ 𝑐𝑒 (𝑡) 𝑑𝑡
𝑒∈𝐸 0
satisfies condition (5) and is therefore a potential. Since the cost functions 𝑐𝑒 are strictly
increasing, 𝛷 is strictly concave, and its restriction to the strategy space 𝑋 therefore has a
unique maximum point. If the marginal social costs are strictly increasing in [0,1], then the
mean payoff 𝜙 is also strictly concave. With 0 ≤ 𝑟 ≤ 1, the same then holds for 𝛷 𝑟 =
(1 − 𝑟)𝛷 + 𝑟𝜙, which as indicated (see Section ‎2.1) is a potential for 𝑔𝑟 . By the remarks
that follow Proposition 2, the unique maximum point of 𝛷 𝑟 is the unique equilibrium
strategy in 𝑔𝑟 . This conclusion and Proposition 2 itself give the following result.
Proposition 6. Consider a nonatomic congestion game where for each resource 𝑒 the
marginal social cost 𝑀𝐶𝑒 is strictly increasing in [0,1]. For the corresponding population
game 𝑔, with strategy space 𝑋, and the mean payoff 𝜙 as the social payoff, the modified
game 𝑔𝑟 has a unique equilibrium strategy 𝑦 𝑟 for every 0 ≤ 𝑟 ≤ 1. For 0 ≤ 𝑟 < 𝑠 ≤ 1,
𝜙(𝑦 𝑟 ) ≤ 𝜙(𝑦 𝑠 ) ≤ 𝜙(𝑦1 ) = max 𝜙(𝑥).
𝑥∈𝑋
Proposition 6 shows that nonatomic congestion games, like the linear congestion games
with splittable flow studied in Section ‎4, always exhibit “normal” 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 𝑒 the cost
function 𝑐𝑒 is a polynomial and 𝑀𝐶𝑒 is strictly increasing in [0,1]. For the corresponding
20
population game 𝑔, and the mean payoff 𝜙 as the social payoff, the function that assigns to
each 0 ≤ 𝑟 ≤ 1 the unique equilibrium strategy 𝑦 𝑟 in the modified game 𝑔𝑟 is continuous.
The function 𝜋: [0,1] → ℝ defined by
𝜋(𝑟) = 𝜙(𝑦 𝑟 )
is absolutely continuous and there is a partition of [0,1] into finitely many intervals such that
𝜋 is either constant or strictly increasing in each interval.
Proof. In view of Proposition 6, the graph of the function 𝜋 is the projection on the first two
coordinates of the set
𝒜 = { (𝑟, 𝛼, 𝑦) ∣ 0 ≤ 𝑟 ≤ 1, 𝛼 = 𝜙(𝑦), 𝑦 is an equilibrium strategy in 𝑔𝑟 }.
The equilibrium condition for 𝑦 is expressed by (3). Since 𝑔𝑟 (𝑥, 𝑦) is linear in the first
argument 𝑥 (see (21)), the condition holds if and only if the inequality in (3) holds for every 𝑥
that is a pure strategy. As there are only finitely many pure strategies, 𝒜 is a semialgebraic
compact set. It follows that 𝜋 is continuous and semialgebraic and that the function 𝑟 ↦ 𝑦 𝑟
is continuous. The other assertions concerning 𝜋 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.
∎
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
𝑒 the cost function 𝑐𝑒 is a polynomial with nonnegative coefficients. For the corresponding
family of population games 𝒢, as well as every subfamily or individual game, the price of
anarchy PoA𝑟 is nonincreasing for 0 ≤ 𝑟 ≤ 1, with 1 ≤ PoA𝑟 ≤ 1/𝑟.
The bound PoA𝑟 ≤ 1/𝑟, 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 𝑘 ≥ 1, the price of anarchy can
be determined precisely. For 0 ≤ 𝑟 ≤ 1,
1
−1
1 + 𝑟𝑘 1+𝑘
𝑟
) ) .
PoA = (1 + 𝑟𝑘 − 𝑘 (
1+𝑘
As 𝑘 increases, this expression monotonically increases and tends to the limit (hence, the
least upper bound) (𝑟(1 − ln 𝑟))
al. 2014 mistakenly assert).
−1
(and not to 1/𝑟, 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, “positive” comparative statics, whereby
an increase in the altruism coefficient increases social welfare, and (definite) instability is
21
associated with “negative” 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 players’ 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 players’ common strategy space 𝑋.9 In principle, any topology may
be used, but in many games, there is a single natural one.
Definition 3. A strategy 𝑦 in a symmetric 𝑁-player game with payoff function 𝑔 is stable,
weakly stable or definitely unstable if it has a neighborhood where, for every strategy 𝑥 ≠ 𝑦,
the inequality
𝑁
1
∑(𝑔(𝑥, 𝑥,
⏟… , 𝑥 , 𝑦,
⏟… , 𝑥 , 𝑦,
⏟… , 𝑦 ) − 𝑔(𝑦, 𝑥,
⏟… , 𝑦 )) < 0,
𝑁
𝑗−1 times
𝑗−1 times
𝑗=1
𝑁−𝑗 times
(23)
𝑁−𝑗 times
a similar weak inequality or the reverse (strict) inequality, respectively, holds. If the
corresponding inequality holds for all strategies 𝑥 ≠ 𝑦,10 then 𝑦 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 𝑦 to the nearby
strategy 𝑥 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 𝑦 in a population game 𝑔 is stable, weakly stable or definitely
unstable if it has a neighborhood where, for every strategy 𝑥 ≠ 𝑦, the inequality
1
∫ (𝑔(𝑥, 𝑝𝑥 + (1 − 𝑝)𝑦) − 𝑔(𝑦, 𝑝𝑥 + (1 − 𝑝)𝑦)) 𝑑𝑝 < 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, 𝑁-player games where the strategy space of each player 𝑖 is
endowed with a specified topology. This is because any 𝑁-player game ℎ can be
symmetrized by letting the players switch roles, with all possible permutations considered.
Thus, in the symmetric 𝑁-player game 𝑔 obtained by symmetrizing ℎ, the players’ common
9
This notion of (topological) locality should not be confused with the one used in Section ‎3, 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 𝑋, where the only neighborhood of
any strategy is the entire strategy set.
22
strategy space is the product space 𝑋 = 𝑋1 × 𝑋2 × ⋯ × 𝑋𝑁 , whose elements are the
strategy profiles in the original, asymmetric game and the topology is the product topology.
For a player in 𝑔, a strategy 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑁 ) ∈ 𝑋 determines the strategy 𝑥𝑖 the player
will use when called to assume the role of any player 𝑖 in ℎ. All 𝑁! assignments of players in
𝑔 to roles in ℎ are considered, and a player’s payoff is defined as his average payoff in these
1 ), 2
assignments. Formally, for any 𝑁 strategies in 𝑋, 𝑥 1 = (𝑥11 , 𝑥21 , … , 𝑥𝑁
𝑥 = (𝑥12 , 𝑥22 , … , 𝑥𝑁2 ),
… , 𝑥 𝑁 = (𝑥1𝑁 , 𝑥2𝑁 , … , 𝑥𝑁𝑁 ),
𝑔(𝑥 1 , 𝑥 2 , … , 𝑥 𝑁 ) =
1
𝜌(1)
𝜌(2)
𝜌(𝑁)
∑ ℎ𝜌−1 (1) (𝑥1 , 𝑥2 , … , 𝑥𝑁 ),
𝑁!
(24)
𝜌∈Π
where Π is the set of all permutation of (1,2, … , 𝑁). (Note that superscripts in this formula
index players’ strategies in the symmetric game while subscripts refer to roles in the
asymmetric one. For 𝜌 ∈ Π, 𝜌(𝑖) is the player assigned to role 𝑖.) A strategy profile
𝑦 = (𝑦1 , 𝑦2 , … , 𝑦𝑁 ) in ℎ may be viewed as stable, weakly stable or definitely unstable if it
has the same property as a strategy in 𝑔. The following result expresses these conditions
directly in terms of payoffs in the asymmetric game.
Lemma 4. A strategy profile 𝑦 in an 𝑁-player game ℎ is stable, weakly stable or definitely
unstable as a strategy in the game obtained by symmetrizing ℎ if and only if it has a
neighborhood where, for every strategy profile 𝑥 ≠ 𝑦, the inequality
∑
1
𝑁
𝑆 (|𝑆|)
(ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ̅𝑆 ∁ ( 𝑦 ∣ 𝑥𝑆 )) < 0,
(25)
a similar weak inequality or the reverse (strict) inequality, respectively, holds.
The summation in (25) is over all sets of players 𝑆 ⊆ {1,2, … , 𝑁}. For each 𝑆, ℎ̅𝑆 =
(1/|𝑆|) ∑𝑖∈𝑆 ℎ𝑖 (= 0, by convension, if 𝑆 = ∅) is the average payoff of the players in 𝑆, ℎ̅𝑆 ∁ is
the average for the complementary set 𝑆 ∁ , and 𝑦 ∣ 𝑥𝑆 denotes the strategy profile where the
players in and outside 𝑆 play according to 𝑥 and 𝑦, respectively. Roughly, inequality (25) says
that the 𝑥 players fare worse on average than the 𝑦 players.
Proof. Since the game 𝑔 obtained by symmetrizing ℎ is invariant to permutations of its
second through 𝑁th 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
∑
1
𝑁−1
𝑇 (|𝑇|−1)
(𝑔(𝑥,
⏟… , 𝑥 , 𝑦,
⏟… , 𝑥 )).
⏟… , 𝑦 ) − 𝑔(𝑦,
⏟… , 𝑦 , 𝑥,
1∈𝑇
|𝑇| times 𝑁−|𝑇| times
(26)
|𝑇| times 𝑁−|𝑇| times
The expression in parenthesis is the difference between the payoff of each player in 𝑇 when
everyone in this set plays 𝑥 and the players outside it play 𝑦 and the payoff when the
strategies are interchanged. Therefore, by (24), the sum (26) is equal to
1
1
(ℎ𝜌−1 (1) ( 𝑦 ∣ 𝑥𝜌−1 (𝑇) ) − ℎ𝜌−1 (1) ( 𝑥 ∣ 𝑦𝜌−1 (𝑇) )).
∑∑
𝑁−1
𝑁!
)
𝑇 (
𝜌∈Π
1∈𝑇
(27)
|𝑇|−1
The expression in parenthesis has the form ℎ𝑖 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ𝑖 ( 𝑥 ∣ 𝑦𝑆 ), with 𝑖 ∈ 𝑆. Specifically,
23
𝑖 = 𝜌−1 (1) and 𝑆 = 𝜌−1 (𝑇). For every pair (𝑆, 𝑖) with 𝑖 ∈ 𝑆 there are precisely (𝑁 − 1)!
pairs (𝜌, 𝑇) satisfying the last pair of equations (and, automatically, 1 ∈ 𝑇 and |𝑇| = |𝑆|).
Therefore, (27) is equal to
(𝑁 − 1)!
1
1
(ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ̅𝑆 ( 𝑥 ∣ 𝑦𝑆 )).
∑∑
(ℎ𝑖 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ𝑖 ( 𝑥 ∣ 𝑦𝑆 )) = ∑
𝑁!
( 𝑁−1 )
(𝑁)
𝑆≠∅ 𝑖∈𝑆
𝑆
|𝑆|−1
|𝑆|
Changing the summation variable in the expression ℎ̅𝑆 ( 𝑥 ∣ 𝑦𝑆 ) from 𝑆 to 𝑆 ∁ gives the lefthand side of (25).
∎
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 𝑃 is a potential for an 𝑁-player ℎ, where the set of all strategy profiles is 𝑋, then
a potential for the symmetric game obtained by symmetrizing ℎ is the symmetric function
𝐹: 𝑋 𝑁 → ℝ defined by
𝐹(𝑥 1 , 𝑥 2 , … , 𝑥 𝑁 ) =
1
𝜌(1)
𝜌(2)
𝜌(𝑁)
∑ 𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑁 ).
𝑁!
(28)
𝜌∈Π
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 𝑦 in an 𝑁-player game ℎ with a potential 𝑃 is stable, weakly
stable or definitely unstable as a strategy in the symmetric game 𝑔 obtained by
symmetrizing ℎ if and only if 𝑦 is a strict local maximum point, local maximum point or strict
local minimum point of 𝑃, respectively. A global maximum point of 𝑃 is both globally weakly
stable (and if it is a strict global maximum point, globally stable) in 𝑔 and an equilibrium in ℎ.
Proof. Consider the potential 𝐹 for 𝑔 specified by (28). Since 𝐹 is symmetric (that is, invariant
to permutations of its 𝑁 arguments), the left-hand site of (23) is equal to
𝑁
1
1
∑(𝐹(𝑥,
⏟… , 𝑥 , 𝑦, … , 𝑦) − 𝐹(𝑥,
⏟… , 𝑥 , 𝑦, … , 𝑦)) = (𝐹(𝑥, 𝑥, … 𝑥) − 𝐹(𝑦, 𝑦, … 𝑦)).
𝑁
𝑁
𝑗 times
𝑗−1 times
𝑗=1
Therefore, by Definition 3, 𝑦 is stable, weakly stable or definitely unstable in 𝑔 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 𝑥 ↦ 𝐹(𝑥, 𝑥, … 𝑥). By (28), 𝐹(𝑥, 𝑥, … 𝑥) = 𝑃(𝑥), which implies
that these conditions on 𝐹 are equivalent to those specified for 𝑃. The special case where
the topology is the trivial one gives the last part of the theorem.
∎
A parallel result applies to strategies in population games.
Theorem 5 (Milchtaich 2016). A strategy 𝑦 in a population game 𝑔 with a potential 𝛷 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 𝛷, respectively. In the first two cases,
𝑦 is in addition an equilibrium strategy. If the potential 𝛷 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 𝑟
and 𝑠, 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 𝑟
and the corresponding strategies or strategy profiles, and may be thought of as tracing the
players’ evolving behavior as they respond to the changing 𝑟. Note that, like Propositions 1
and 2, the three theorems hold for any choice of social payoff functions.
Theorem 6. For an 𝑁-player game ℎ, a social payoff 𝑓, and altruism coefficients 𝑟 and 𝑠 with
𝑟 < 𝑠 ≤ 1, if two distinct strategy profiles 𝑦 𝑟 and 𝑦 𝑠 are globally weakly stable as strategies
in the symmetric games obtained by symmetrizing the modified games ℎ𝑟 and ℎ 𝑠 ,
respectively, then
𝑓(𝑦 𝑟 ) ≤ 𝑓(𝑦 𝑠 ).
If the strategies are globally stable, then a similar strict inequality holds.
Proof. The proof uses the following identity, which holds for all (𝑟, 𝑠 and) 𝑥 and 𝑦:
(1 − 𝑠) ∑
1
𝑁
𝑆 (|𝑆|)
(ℎ̅𝑆𝑟 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ̅𝑆𝑟∁ ( 𝑦 ∣ 𝑥𝑆 )) + (1 − 𝑟) ∑
1
𝑁
𝑆 (|𝑆|)
= (1 − 𝑠)(1 − 𝑟) ∑
𝑆
1
𝑁
(|𝑆|
)
(ℎ̅𝑆𝑠 ( 𝑥 ∣ 𝑦𝑆 ) − ℎ̅𝑆𝑠∁ ( 𝑥 ∣ 𝑦𝑆 ))
((ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ) − ℎ̅𝑆 ∁ ( 𝑦 ∣ 𝑥𝑆 )) + (ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ∁ ) − ℎ̅𝑆 ∁ ( 𝑦 ∣ 𝑥𝑆 ∁ )))
+(1 − 𝑠)𝑟 (𝑓(𝑥) − 𝑓(𝑦)) + (1 − 𝑟)𝑠 (𝑓(𝑦) − 𝑓(𝑥)) = (𝑟 − 𝑠)(𝑓(𝑥) − 𝑓(𝑦)),
where the second equality follows from the fact that, on its left-hand side, the coefficients of
ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ) and ℎ̅𝑆 ( 𝑦 ∣ 𝑥𝑆 ∁ ) vanish for every set of players 𝑆. By Lemma 4, applied to the
trivial topology (see footnote 10), the two sums in the identity are nonpositive for 𝑦 = 𝑦 𝑟
and 𝑥 = 𝑦 𝑠 if these strategy profiles are globally weakly stable as strategies in the symmetric
games obtained by symmetrizing the modified games ℎ𝑟 and ℎ 𝑠 , respectively, and they are
negative if the strategies are globally stable.
∎
Theorem 7 (Milchtaich 2012). For an 𝑁-player game ℎ and a social payoff function 𝑓 that are
both Borel measurable11, and altruism coefficients 𝑟0 and 𝑟1 with 𝑟0 < 𝑟1 ≤ 1, suppose that
there is a continuous and finitely-many-to-one12 function assigning to each 𝑟0 ≤ 𝑟 ≤ 𝑟1 a
strategy profile 𝑦 𝑟 such that the function 𝜋: [𝑟0 , 𝑟1 ] → ℝ 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
𝜋(𝑟) = 𝑓(𝑦 𝑟 )
(29)
is absolutely continuous.13 If, for every 𝑟0 < 𝑟 < 𝑟1 , the strategy profile 𝑦 𝑟 is stable,
weakly stable or definitely unstable as a strategy in the game obtained by symmetrizing the
modified game ℎ𝑟 , then 𝜋 is strictly increasing, nondecreasing or strictly decreasing,
respectively.
Theorem 8 (Milchtaich 2012). For a population game 𝑔 and a social payoff function 𝜙 that
are both Borel measurable, and altruism coefficients 𝑟0 and 𝑟1 with 𝑟0 < 𝑟1 ≤ 1, suppose
that there is a continuous and finitely-many-to-one function assigning to each 𝑟0 ≤ 𝑟 ≤ 𝑟1 a
strategy 𝑦 𝑟 such that the function 𝜋: [𝑟0 , 𝑟1 ] → ℝ defined by
𝜋(𝑟) = 𝜙(𝑦 𝑟 )
is absolutely continuous. If, for every 𝑟0 < 𝑟 < 𝑟1 , strategy 𝑦 𝑟 is a stable, weakly stable or
definitely unstable in the modified game 𝑔𝑟 , then 𝜋 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 𝑟 ↦ 𝑦 𝑟 that is reasonably
“smooth” (the continuity and absolute continuity conditions) and “responsive” (the finitelymany-to-one condition). In this general setting, they show that social welfare necessary
increases with increasing 𝑟 if each 𝑦 𝑟 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 𝑁-player game ℎ where the strategy space
of each player 𝑖 is a set in a Euclidean space ℝ𝑛𝑖 , for some (possibly, player-specific) 𝑛𝑖 ≥ 1.
A strategy profile 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑁 ) is viewed as an 𝑛-dimensional column vector, where
𝑛 = ∑𝑖 𝑛𝑖 . It is an interior strategy profile if each strategy 𝑥𝑖 lies in the interior of player 𝑖’s
strategy space (equivalently, 𝑥 lies in the interior of the product space). The gradient with
respect to the components of player 𝑖’s strategy is denoted ∇𝑖 and is viewed as an 𝑛𝑖 dimensional row vector (of first-order differential operators). For any 𝑖 and 𝑗, ∇T𝑖 ∇𝑗 is
therefore an 𝑛𝑖 × 𝑛𝑗 matrix (of second-order differential operators). For example, ∇T𝑖 ∇𝑖 ℎ𝑖 is
the Hessian matrix of player 𝑖’s payoff function with respect to the player’s own strategy.
These Hessian matrices are the diagonal blocks in the 𝑛 × 𝑛 block matrix
∇1T ∇1 ℎ1
𝐻=(
⋮
∇T𝑁 ∇1 ℎ𝑁
13
⋯
⋱
⋯
∇1T ∇𝑁 ℎ1
).
⋮
∇T𝑁 ∇𝑁 ℎ𝑁
A sufficient condition for absolute continuity is that the function is continuously differentiable.
26
(30)
The value that the matrix 𝐻 attains when its entries are evaluated at a strategy profile 𝑥 is
denoted 𝐻(𝑥).
Proposition 7. In an 𝑁-player game ℎ where the strategy space of each player is a set in a
Euclidean space, let 𝑦 be an interior equilibrium that has a neighborhood where the players’
payoff functions have continuous second-order derivatives. A sufficient condition for 𝑦 to be
stable or definitely unstable as a strategy in the game obtained by symmetrizing ℎ is that the
symmetric matrix (1/2)(𝐻(𝑦) + 𝐻(𝑦)T ) is negative definite or positive definite,
respectively.14 A necessary condition for weak stability is that the matrix is negative
semidefinite.
Proof. With 𝜒𝑆 denoting the characteristic function of a set of players 𝑆, the left-hand side of
(25) can be written as
1
1
(𝜒𝑆 (𝑖) ℎ𝑖 ( 𝑦 ∣ 𝑥𝑆 ) − 𝜒𝑆 ∁ (𝑖) ℎ𝑖 ( 𝑦 ∣ 𝑥𝑆 )).
∑∑
𝑁−1
𝑁
𝑆
𝑖 (|𝑆∖{𝑖}|)
(31)
For 𝑥 tending to 𝑦, equivalently, ϵ𝑖 = 𝑥𝑖 − 𝑦𝑖 → 0 for all 𝑖, this expression can be written as
1
1
1
(𝜒𝑆 (𝑖) − 𝜒𝑆 ∁ (𝑖)) (ℎ𝑖 + ∑ ∇𝑗 ℎ𝑖 ϵ𝑗 + ∑ ∑ ϵT𝑘 ∇T𝑘 ∇𝑗 ℎ𝑖 ϵ𝑗 ) + 𝑜(‖𝜖‖2 ),
∑∑
𝑁−1
𝑁
2
𝑆
𝑖 (|𝑆∖{𝑖}|)
𝑗∈𝑆
𝑗∈𝑆 𝑘∈𝑆
(32)
where the payoff function ℎ𝑖 and its partial derivatives are evaluated at the point 𝑦 and ‖𝜖‖
is the (Euclidean) length of the vector ϵ = (ϵ1 , ϵ2 , … , ϵ𝑁 ) = 𝑥 − 𝑦. For each player 𝑖, the
coefficient of ℎ𝑖 in (32) is
1
1
(𝜒𝑆 (𝑖) − 𝜒𝑆 ∁ (𝑖))
∑
𝑁−1
𝑁
(
)
𝑆
|𝑆∖{𝑖}|
=
1
1
[(𝜒𝑆 (𝑖) − 𝜒𝑆 ∁ (𝑖)) + (𝜒𝑆∪{𝑖} (𝑖) − 𝜒(𝑆∪{𝑖})∁ (𝑖))] = 0.
∑
𝑁 𝑆 ( 𝑁−1 )
|𝑆∖{𝑖}|
𝑖∉𝑆
The last equality holds because the condition 𝑖 ∉ 𝑆 implies that the expression in square
brackets is zero. For each 𝑖 and 𝑗, the coefficient of ∇𝑗 ℎ𝑖 ϵ𝑗 in (32) is
1
1
(𝜒 (𝑖) − 𝜒𝑆 ∁ (𝑖)),
∑
𝑁 𝑆 ( 𝑁−1 ) 𝑆
|𝑆∖{𝑖}|
𝑗∈𝑆
which by a similar argument is zero if 𝑗 ≠ 𝑖. For each 𝑖, 𝑗 and 𝑘, the coefficient of ϵT𝑘 ∇T𝑘 ∇𝑗 ℎ𝑖 ϵ𝑗
is
1
1
(𝜒 (𝑖) − 𝜒𝑆 ∁ (𝑖)),
∑
2𝑁 𝑆 ( 𝑁−1 ) 𝑆
|𝑆∖{𝑖}|
(33)
𝑗,𝑘∈𝑆
14
The negative definiteness condition implies that 𝐻(𝑦) 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 𝑗 and 𝑘 are both different from 𝑖. If either 𝑗 ≠ 𝑖 and 𝑘 = 𝑖 or 𝑗 = 𝑖 and
𝑘 ≠ 𝑖, then (33) is equal to
𝑁
𝑁
𝑙=2
𝑙=2
(𝑁−2
)
1
1
1
1
𝑙−1
1
𝑙−2
∑
=
∑ 𝑁−1
=
∑
= .
2𝑁 𝑆 ( 𝑁−1 ) 2𝑁
𝑁−1 4
( 𝑙−1 ) 2𝑁
𝑗,𝑘∈𝑆
|𝑆|−1
If 𝑗 = 𝑘 = 𝑖, then (33) is equal to
𝑁
(𝑁−1
) 1
1
1
1
𝑙−1
∑
=
∑ 𝑁−1
= .
2𝑁 𝑆 ( 𝑁−1 ) 2𝑁
( 𝑙−1 ) 2
𝑖∈𝑆
𝑙=1
|𝑆|−1
Finally, ∇𝑖 ℎ𝑖 = 0 for each player 𝑖, because the interior equilibrium 𝑦 necessarily satisfies
these first-order maximization conditions. Therefore, (32) reads
(34)
1
1
∑ ( ∑ ϵT𝑖 ∇T𝑖 ∇𝑗 ℎ𝑖 ϵ𝑗 + ∑ ϵT𝑘 ∇T𝑘 ∇𝑖 ℎ𝑖 ϵ𝑖 ) + 𝑜(‖𝜖‖2 )
4
4
𝑖
𝑗
𝑘
1
1
= ϵT (𝐻(𝑦) + 𝐻(𝑦)T )ϵ + 𝑜(‖𝜖‖2 ) = ϵT 𝐻(𝑦)ϵ + 𝑜(‖𝜖‖2 ).
4
2
If (1/2)(𝐻(𝑦) + 𝐻(𝑦)T ) is negative definite or positive definite and ϵ ≠ 0, then ϵT 𝐻ϵ is
negative or positive, respectively, and its absolute value is at least |𝜆0 |‖𝜖‖2 , where 𝜆0 ≠ 0 is
the eigenvalue closest to 0 of (1/2)(𝐻(𝑦) + 𝐻(𝑦)T ). Therefore, in the first or second case,
(34) is positive or negative for 𝜖 sufficiently close to 0, which proves that (25) or the reverse
inequality, respectively, holds for 𝑥 sufficiently close to 𝑦.
If (1/2)(𝐻(𝑦) + 𝐻(𝑦)T ) is not negative semidefinite, then it has an eigenvector 𝜂 with
eigenvalue 𝜆 > 0, so that 𝜂T 𝐻(𝑦)𝜂 is positive and equal to 𝜆‖𝜂‖2 . It follows that there are
strategy profiles 𝑥 arbitrarily close to 𝑦 for which the reverse of (25) holds, which means that
𝑦 is not weakly stable.
∎
Proposition 7 may be applied also to an interior equilibrium 𝑦 𝑟 in the modified game ℎ𝑟 , for
any altruism coefficient 𝑟 and any social payoff function 𝑓 that has continuous second-order
partial derivatives in a neighborhood of 𝑦 𝑟 . Here, the relevant matrix is 𝐻𝑟 , which is
obtained from 𝐻 be replacing ℎ with ℎ𝑟 . This observation suggests that, if the matrix
(1/2)(𝐻𝑟 (𝑦 𝑟 ) + 𝐻𝑟 (𝑦 𝑟 )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 ℎ, there are 𝑁 ≥ 3 players and three
resources. Player 1 has weight 𝑤1 = 1 and can only use resources 𝑒 and 𝑒 ′ . Each of the
other players 𝑖 has weight 𝑤𝑖 = 1/(𝑁 − 1) and can only use resources 𝑒 and 𝑒 ″ . The (perunit) costs for 𝑒 ′ and 𝑒 ″ are constant, 𝑏 ′ and 𝑏 ″ respectively, and that of 𝑒 is given by a
continuous cost function 𝑐𝑒 that is twice continuously differentiable outside the origin, with
𝑐𝑒′ (𝑡) > 0 and (𝑀𝐶𝑒 )′ (𝑡) = 2𝑐𝑒′ (𝑡) + 𝑡 𝑐𝑒″ (𝑡) ≥ 0,
𝑡 > 0.
(The marginal social cost 𝑀𝐶𝑒 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
𝑐𝑒 (𝑡) = 𝑡 𝑑 .
(36)
For example, with 𝑑 = 1, 𝑏 ′ = 2.8 and 𝑏 ″ = 1.9, they show that if the number of
players 𝑁 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 𝑟.
Claim. Suppose that the cost function 𝑐𝑒 satisfies
𝑡 𝑐𝑒″ (𝑡)
𝑁+1
,
′ (𝑡) < 2
𝑐𝑒
𝑁−1
𝑡 > 0.
If ℎ has an equilibrium where all players use both their resources, then the modified game
ℎ𝑟 has a unique such equilibrium 𝑦 𝑟 for all 𝑟 in some interval [0, 𝑟1 ], with 0 < 𝑟1 < 1, and
the corresponding aggregate payoff 𝑓(𝑦 𝑟 ) is strictly increasing (and the aggregate cost is
strictly decreasing) in this interval.
For the power function (36), the above condition on 𝑐𝑒 (together with (35)) translates to
0<𝑑<
3𝑁 + 1
.
𝑁−1
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 ℎ.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)(𝐻 + 𝐻T ) (or the positive
definiteness of its negative).
Note that, with 𝑐𝑒 given by a power function, ℎ is a potential game only in the linear case,
𝑑 = 1.
To prove the Claim, note first that, since each player 𝑖 can only use two resources, the
player’s strategy 𝑥𝑖 is completely specified by the component 𝑥𝑖𝑒 . It is therefore possible to
identify 𝑥𝑖 with 𝑥𝑖𝑒 , so that the strategy space becomes unidimensional, specifically, the
interval [0, 𝑤𝑖 ]. Thus, a strategy profile has the form 𝑥 = (𝑥1𝑒 , 𝑥2𝑒 , … , 𝑥𝑁𝑒 ). For 0 ≤ 𝑟 ≤ 1,
the derivative of player 1’s payoff in the modified game ℎ𝑟 with respect to the player’s
strategy, denoted (ℎ1𝑟 )1 , at a point 𝑥 ≠ 0 is given by
−(ℎ1𝑟 )1 (𝑥) = 𝑐𝑒 (𝑥𝑒 ) + ((1 − 𝑟)𝑥1𝑒 + 𝑟𝑥𝑒 )𝑐𝑒′ (𝑥𝑒 ) − 𝑏 ′ ,
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
−(ℎ1𝑟 )11 (𝑥) = 2𝑐𝑒′ (𝑥𝑒 ) + ((1 − 𝑟)𝑥1𝑒 + 𝑟𝑥𝑒 )𝑐𝑒″ (𝑥𝑒 )
= 2(1 − 𝑟) (1 −
𝑥1𝑒 ′
𝑥1𝑒
) 𝑐𝑒 (𝑥𝑒 ) + ((1 − 𝑟)
+ 𝑟) (𝑀𝐶𝑒 )′ (𝑥𝑒 )
𝑥𝑒
𝑥𝑒
≥ 0.
The inequality, which follows from (35), shows that the payoff is concave with respect to
player 1’s own strategy 𝑥1𝑒 . The same conclusion holds for any other player 𝑖, as the only
difference is that 𝑏 ′ in this case is replaced with 𝑏 ″ . Therefore, an interior equilibrium 𝑦 𝑟 in
ℎ𝑟 is completely characterized by the first-order conditions (ℎ𝑖𝑟 )𝑖 (𝑦 𝑟 ) = 0 (𝑖 = 1,2, … , 𝑁).
Summation of these equations shows that 𝑦 𝑟 must be a solution of
𝑁𝑐𝑒 (𝑥𝑒 ) + (1 + 𝑟(𝑁 − 1)) 𝑥𝑒 𝑐𝑒′ (𝑥𝑒 ) = 𝑏 ′ + (𝑁 − 1)𝑏 ″ .
(37)
The expression on the left-hand side can be written as
(1 − 𝑟)(𝑁 − 1) 𝑐𝑒 (𝑥𝑒 ) + (1 + 𝑟(𝑁 − 1)) 𝑀𝐶𝑒 (𝑥𝑒 ).
By (35), for 0 ≤ 𝑟 < 1, the derivative of this expression with respect to 𝑥𝑒 and the derivative
with respect to 𝑟 are both positive if 𝑥𝑒 > 0. Therefore, for such 𝑟, a positive solution to (37)
is unique and, by the implicit function theorem, is determined by 𝑟 as a continuously
differentiable function with a negative derivative. Thus, if 𝑦 0 is an interior equilibrium in the
unmodified game ℎ, there exists some 0 < 𝑟1 < 1 and a continuously differentiable
mapping 𝑟 ↦ 𝑦𝑒𝑟 on [0, 𝑟1 ] with 𝑑𝑦𝑒𝑟 / 𝑑𝑟 < 0 such that 𝑦𝑒𝑟 solves (37). Setting 𝑥𝑒 = 𝑦𝑒𝑟 in
each of the first-order conditions (ℎ𝑖𝑟 )𝑖 = 0 now gives a linear equation for 𝑥𝑖𝑒 . If 𝑟1 is
𝑟
𝑟
sufficiently close to 0, the solution 𝑦𝑖𝑒
satisfies 0 < 𝑦𝑖𝑒
< 𝑤𝑖 for every 0 ≤ 𝑟 ≤ 𝑟1 , and the
𝑟
𝑟
𝑟
𝑟
derivative 𝑑𝑦𝑖𝑒 / 𝑑𝑟 exists and is continuous. (Necessarily, 𝑦2𝑒
= 𝑦3𝑒
= ⋯ = 𝑦𝑁𝑒
, as the
corresponding 𝑁 − 1 equations are identical.) Summation of the 𝑁 equations and
𝑟
comparison with (37) show that 𝑦𝑖𝑒
necessarily sum up to 𝑦𝑒𝑟 , so that they specify the unique
interior equilibrium in ℎ𝑟 . In addition, since 𝑑𝑦𝑒𝑟 / 𝑑𝑟 < 0, the function 𝑟 ↦ 𝑦 𝑟 is one-to-one
on [0, 𝑟1 ] and 𝑑𝑦 𝑟 / 𝑑𝑟 ≠ 0 there.
By the last conclusion, Proposition 7 and Theorem 7, a sufficient condition for 𝑟 ↦ 𝑦 𝑟 to be
strictly increasing in [0, 𝑟1 ] is that the matrix (1/2)(𝐻𝑟 (𝑦 𝑟 ) + 𝐻𝑟 (𝑦 𝑟 )T ) is positive definite
𝑁
for 0 < 𝑟 < 𝑟1 . The matrix 𝐻𝑟 = ((ℎ𝑖𝑟 )𝑖𝑗 )
𝑖,𝑗=1
is explicitly given by the following equations.
For all 𝑖 and 𝑗 ≠ 𝑖,
−(ℎ𝑖𝑟 )𝑖𝑖 (𝑥) = 2𝑐𝑒′ (𝑥𝑒 ) + ((1 − 𝑟)𝑥𝑖𝑒 + 𝑟𝑥𝑒 )𝑐𝑒″ (𝑥𝑒 ),
−(ℎ𝑖𝑟 )𝑖𝑗 (𝑥) = (1 + 𝑟)𝑐𝑒′ (𝑥𝑒 ) + ((1 − 𝑟)𝑥𝑗𝑒 + 𝑟𝑥𝑒 ) 𝑐𝑒″ (𝑥𝑒 ).
Therefore,
−(1/2)(𝐻𝑟 (𝑦 𝑟 ) + 𝐻𝑟 (𝑦 𝑟 )T ) = (1 − 𝑟) (𝛼𝐼 + 𝛽𝐸 + 𝛾(𝐹 + 𝐹 T )) + 𝑟𝛿𝐸,
(38)
where 𝐼 is the identity matrix, 𝐸 is the matrix with all entries 1, 𝐹 has 1’s in the first row and
0’s elsewhere,
30
𝛼
𝛽
𝛾
𝛿
= 𝑐𝑒′ (𝑦𝑒𝑟 )
𝑟 ″ (𝑦 𝑟 )
= 𝑐𝑒′ (𝑦𝑒𝑟 ) + 𝑦2𝑒
𝑐𝑒 𝑒
𝑟
𝑟 )𝑐 ″ (𝑦 𝑟 )
= (1/2)(𝑦1𝑒 − 𝑦2𝑒
𝑒 𝑒
= 2𝑐𝑒′ (𝑦𝑒𝑟 ) + 𝑦𝑒𝑟 𝑐𝑒″ (𝑦𝑒𝑟 ).
𝑟
𝑟 )/(𝑁
Note that, since 𝑦2𝑒
= (𝑦𝑒𝑟 − 𝑦1𝑒
− 1) < 𝑦𝑒𝑟 /2, it follows from (35) that the
expressions 𝛼, 𝛽, 𝛽 + 𝛾 and 𝛿 are positive. If 𝛾 = 0, then, since 𝐸 is positive semidefinite,
the right-hand side of (38) is positive definite. If 𝛾 ≠ 0, then to reach the same conclusion it
suffices to show that all the eigenvalues of the matrix 𝛽𝐸 + 𝛾(𝐹 + 𝐹 T ) are greater than −𝛼.
The rank of this 𝑛 × 𝑛 matrix is 2. Its two non-zero eigenvalues are the roots of the quadratic
polynomial
𝑄(𝑠) = 𝑠 2 − (𝑁𝛽 + 2𝛾)𝑠 − (𝑁 − 1)𝛾 2 ,
because it is not difficult to check that the condition for ((𝑠 − (𝑁 − 1)𝛽)/(𝛽 + 𝛾), 1,1, … ,1)
to be an eigenvector with eigenvalue 𝑠 is 𝑄(𝑠) = 0. Since 𝑄(0) < 0, the negative number
−𝛼 is smaller than both roots of 𝑄 if and only if 𝑄(−𝛼) > 0. Since 𝑁𝛽 + 2𝛾 = 𝑁𝑐𝑒′ (𝑦𝑒𝑟 ) +
𝑦𝑒𝑟 𝑐𝑒″ (𝑦𝑒𝑟 ) and |𝛾| ≤ (1/2)|𝑦𝑒𝑟 𝑐𝑒″ (𝑦𝑒𝑟 )|, a sufficient condition for 𝑄(−𝛼) > 0 is that, for all
𝑡 > 0,
2
2
(𝑐𝑒′ (𝑡)) − (𝑁𝑐𝑒′ (𝑡) + 𝑡𝑐𝑒″ (𝑡))(−𝑐𝑒′ (𝑡)) − (1/4)(𝑁 − 1)(𝑡𝑐𝑒″ (𝑡)) > 0.
The solution of this quadratic inequality is
−2 <
𝑡𝑐𝑒″ (𝑡)
𝑁+1
.
′ (𝑡) < 2
𝑐𝑒
𝑁−1
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)(𝐻𝑟 (𝑦 𝑟 ) + 𝐻𝑟 (𝑦 𝑟 )T ) (equivalently, of the quadratic form corresponding to
𝐻𝑟 (𝑦 𝑟 )) with the effect of altruism and spite on the social payoff.
Proposition 8. For an 𝑁-player game ℎ where the strategy space of each player is a set in a
Euclidean space, a social payoff function 𝑓, and altruism coefficients 𝑟0 and 𝑟1 with 𝑟0 < 𝑟1 ,
suppose that there is a continuously differentiable function assigning to each 𝑟0 < 𝑟 < 𝑟1 an
interior equilibrium 𝑦 𝑟 in the modified game ℎ𝑟 that has a neighborhood where the players’
payoff functions and 𝑓 have continuous second-order derivatives. At each point 𝑟0 < 𝑟 < 𝑟1 ,
T
𝑑
𝑑𝑦 𝑟
𝑑𝑦 𝑟
𝑓(𝑦 𝑟 ) = −(1 − 𝑟) (
) 𝐻𝑟 (𝑦 𝑟 ) .
𝑑𝑟
𝑑𝑟
𝑑𝑟
Proof. Since 𝑦 𝑟 is an interior equilibrium in ℎ𝑟 , it satisfies the necessary first-order
maximization condition for each player 𝑖, which is
∇𝑖 ℎ𝑖𝑟 (𝑦 𝑟 ) = ∇𝑖 ((1 − 𝑟)ℎ𝑖 + 𝑟𝑓)(𝑦 𝑟 ) = 0.
Total derivation with respect to 𝑟 gives
31
(39)
𝑁
T
𝑑𝑦𝑗𝑟
𝑑
∇𝑖 ℎ𝑖𝑟 (𝑦 𝑟 ) = ∇𝑖 (−ℎ𝑖 + 𝑓)(𝑦 𝑟 ) + ∑ (
) ∇𝑗T ∇𝑖 ℎ𝑖𝑟 (𝑦 𝑟 ) = 0.
𝑑𝑟
𝑑𝑟
𝑗=1
Post-multiplication of both sides of the right equality with (1 − 𝑟) 𝑑𝑦𝑖𝑟 / 𝑑𝑟 and summation
over 𝑖 give
𝑁
∑ ∇𝑖 (−(1 − 𝑟)ℎ𝑖 + (1 − 𝑟)𝑓)(𝑦
T
𝑟)
𝑖=1
𝑟
𝑑𝑦𝑖𝑟
𝑑𝑦 𝑟
T 𝑑𝑦
+ (1 − 𝑟) (
) (𝐻𝑟 (𝑦 𝑟 )) = 0.
𝑑𝑟
𝑑𝑟
𝑑𝑟
By the second equality in (39), the first term above (the sum) is equal to 𝑑𝑓(𝑦 𝑟 )/ 𝑑𝑟.
∎
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 people’s 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 𝑖 is measured by the effect on 𝑖’s own
fitness plus 𝑟 times that of each of the other individuals 𝑗, where 𝑟 is the coefficient of
relatedness between 𝑖 and 𝑗 (Milchtaich 2006a). In a social context, a similar parameter 𝑟
may express individual 𝑖’s level of empathy towards 𝑗. Edgeworth (1881) calls this parameter
the coefficient of effective sympathy. More generally, the altruism coefficient 𝑟 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 individuals’ personal,
material payoffs but is a function of everyone’s 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 𝑟, 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 𝑟. 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 𝑟. 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 𝑟 would not affect the preferences expressed by it. In other words, only the ratio
(1 − 𝑟): 𝑟, 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 𝑟 may be a good approximation. Many of the
phenomena described in this paper are evident already for arbitrarily small 𝑟.
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 𝑟 increases, the price of anarchy in the game may (weakly) increase. The main
results in Section ‎3 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 game’s potential function are
equilibria. The monotonicity property only concerns the “congestion game form”; 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 𝑟 and is nondecreasing in
[−1,1]. That is, an increasingly positive or negative 𝑟 can only increase or decrease social
welfare, respectively, or leave it unchanged. The highest aggregate payoff is attained with
complete selflessness, 𝑟 = 1.
Internalization of social payoff may occur even if a single person’s 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 𝑟 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 𝑟 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 𝑟 = 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 ‎6 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 “positive” or “negative” comparative statics, in the
sense of the effect of increasing altruism of the social payoff. Moreover, the last connection
holds not only for the aggregate or mean payoff but for general social payoff functions. As
demonstrated, it can be used for determining the effect of altruism and spite in congestion
games to which the previous results do not apply.
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35
Appendix
A directed or undirected two-terminal network is, respectively, a directed or undirected
multigraph with terminal vertices 𝑜 and 𝑑 that satisfies the “irredundancy” condition: each
edge 𝑒 and each vertex 𝑣 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 𝑜𝑒1 𝑣1 ⋯ 𝑣𝑛−1 𝑒𝑛 𝑑 (𝑛 ≥ 1), that is, one
that begins at the origin vertex 𝑜 and ends at the destination vertex 𝑑. (An alternative,
simpler, notation is 𝑒1 𝑒2 ⋯ 𝑒𝑛 . ) In the directed case, a route is also required to traverse each
of its edges 𝑒 in the direction assigned to 𝑒, that is, the tail and head vertices of 𝑒 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
network’s 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 𝐺
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 𝐺.
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 𝐺 with more
than one edge, and some edge 𝑒 incident with the origin 𝑜. If no other edge is incident with
𝑜, then by the induction hypothesis there exists at least one directed version of the network
obtained from 𝐺 by contracting 𝑒, that is, eliminating the edge and its non-terminal vertex 𝑣
and replacing 𝑣 with 𝑜 as the terminal vertex of all the edges originally incident with 𝑣. Any
such directed version gives a directed version of 𝐺, in which 𝑒 is directed from 𝑜 to 𝑣.
Suppose, then, that 𝑒 is not the only edge incident with 𝑜, and consider the subnetwork 𝐺𝑒
of 𝐺 whose edges and vertices are all those that belong to some route in 𝐺 where the first
edge is 𝑒.
Claim. If a route 𝑟 in 𝐺 includes a non-terminal vertex 𝑢 that is in 𝐺𝑒 , then every edge
(hence, also every vertex) that follows 𝑢 in 𝑟 is also in 𝐺𝑒 .
It clearly suffices to establish this for the edge 𝑒 ′ that immediately follows 𝑢 in 𝑟. Suppose
that 𝑒 ′ is not in 𝐺𝑒 . Let 𝑣 the first vertex that follows 𝑒 ′ in 𝑟 and is in 𝐺𝑒 . All the edges and
vertices in 𝑟 between 𝑢 and 𝑣 are not in 𝐺𝑒 . Adding them to it creates a new subnetwork of
𝐺, since the addition is equivalent to adding to 𝐺𝑒 a single edge with end vertices 𝑢 and 𝑣
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 𝑟 ′ that includes 𝑒 ′ . Necessarily, 𝑟 ′
also includes 𝑒. The conclusion contradicts the assumption that 𝑒 ′ is not in 𝐺𝑒 , and thus
proves the claim.
Consider now the collections of all edges and vertices that are obtained from those in 𝐺 by
(i) eliminating all edges that are in 𝐺𝑒 and (ii) identifying all non-terminal vertices that are in
𝐺𝑒 with 𝑑. It follows from the Claim that these collections constitute a network 𝐺 ′ (with the
terminal vertices 𝑜 and 𝑑). To see this, consider any edge or non-terminal vertex in them and
a route 𝑟 in 𝐺 that includes it. It follows from the Claim that the inclusion still holds if 𝑟 is
36
replaced with its section 𝑟𝑜𝑢 , which is the path obtained from 𝑟 by deleting all edges and
vertices that follow 𝑢, where 𝑢 is the first vertex in 𝑟, other than 𝑜, that is in 𝐺𝑒 . As 𝑢 is one
of the vertices identified with 𝑑, 𝑟𝑜𝑢 is a route in 𝐺 ′ .
By the induction hypothesis, both 𝐺𝑒 and 𝐺 ′ have directed versions, which together specify
directions for all edges in 𝐺. It remains to show that these directions define a directed
version of 𝐺, that is, the irredundancy condition holds. Any edge or vertex that is or is not in
𝐺𝑒 is included in some route 𝑟 in the directed version of 𝐺𝑒 or some route 𝑟 ′ in the directed
version of 𝐺 ′ , respectively. By construction, such routes honor the edges’ directions in 𝐺.
Route 𝑟 is automatically a route also in 𝐺, whereas 𝑟 ′ is a path in 𝐺 that starts at 𝑜, ends at
some vertex 𝑢 in 𝐺𝑒 , and does not include any other vertex or edge that is in 𝐺𝑒 . It is
therefore possible to extend 𝑟 ′ to a route in 𝐺 that honors the edges’ directions by replacing
″
𝑢 with the section 𝑟𝑢𝑑
of some route 𝑟 ″ in the directed version of 𝐺𝑒 that includes 𝑢, where
″
𝑟𝑢𝑑
is obtained from 𝑟 ″ by deleting all the edges and vertices that precede 𝑢.
∎
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 𝐺
defines a directed version of 𝐺 only if it satisfies the condition mentioned in Section ‎3.1,
which is that the direction of each edge 𝑒 coincides with the direction in which some route
in (the undirected network) 𝐺 traverses 𝑒. However, this necessary condition is not
sufficient. For example, an assignment satisfying it may render a non-terminal degree-two
vertex 𝑣 the head vertex of both edges incident with 𝑣, 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 𝐺. 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 𝐺, the following conditions are
equivalent:
(i)
(ii)
(iii)
For each edge 𝑒, all routes in 𝐺 that include 𝑒 traverse it in the same direction.
For every assignment of allowable directions to the edges in 𝐺 such that the
direction of each edge 𝑒 is that in which some route in 𝐺 traverses 𝑒, all routes in
𝐺 are allowable.
𝐺 is series-parallel.
37