# Price of Anarchy in Stochastic Atomic Congestion Games with Affine Costs

We consider an atomic congestion game with stochastic demand in which each player participates in the game with probability p, and incurs no cost with probability 1-p. We assume that p is common knowledge among all players and that players are independent. For congestion games with affine costs, we provide an analytic expression for the price of anarchy as a function of p, which is monotonically increasing and converges to the well-known bound of 5/2 as p→ 1. On the other extreme, for p≤1/4 the bound is constant and equal to 4/3 independently of the game structure and the number of players. We show that these bounds are tight and are attained on routing games with purely linear costs. Additionally, we also obtain tight bounds for the price of stability for all values of p.

## Authors

• 4 publications
• 13 publications
• 7 publications
• 3 publications
• ### Convergence of Large Atomic Congestion Games

We study the convergence of sequences of atomic unsplittable congestion ...
01/09/2020 ∙ by Roberto Cominetti, et al. ∙ 0

• ### The Price of Anarchy in Routing Games as a Function of the Demand

Most of the literature concerning the price of anarchy has focused on th...
07/23/2019 ∙ by Roberto Cominetti, et al. ∙ 0

• ### Tight Bounds for the Price of Anarchy and Stability in Sequential Transportation Games

In this paper, we analyze a transportation game first introduced by Fota...
07/17/2020 ∙ by Francisco J. M. da Silva, et al. ∙ 0

• ### Signaling in Bayesian Network Congestion Games: the Subtle Power of Symmetry

Network congestion games are a well-understood model of multi-agent stra...
02/12/2020 ∙ by Matteo Castiglioni, et al. ∙ 0

• ### The Curse of Ties in Congestion Games with Limited Lookahead

We introduce a novel framework to model limited lookahead in congestion ...
04/19/2018 ∙ by Carla Groenland, et al. ∙ 0

• ### On the Price of Anarchy of Cost-Sharing in Real-Time Scheduling Systems

We study cost-sharing games in real-time scheduling systems where the ac...
07/12/2019 ∙ by Eirini Georgoulaki, et al. ∙ 0

• ### Carrots or Sticks? The Effectiveness of Subsidies and Tolls in Congestion Games

Are rewards or penalties more effective in influencing user behavior? Th...
10/06/2019 ∙ by Bryce L. Ferguson, et al. ∙ 0

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## 1. Introduction

The effects of selfishness in congestion games have been studied extensively in the past twenty years. Initially, the focus was on the simplest case of complete information routing games where either the network was very simple—Koutsoupias and Papadimitriou (1999)—or the demand was infinitely divisible—Roughgarden and Tardos (2002). The impact of selfishness is typically evaluated through worst-case bounds for the price of anarchy (PoA), parameterized by input parameters. Different papers made different calls about what aspect to highlight, hence which parametrization to choose. In Roughgarden and Tardos (2002), it is proved that the PoA for nonatomic congestion games with affine costs is at most . In time, research addressed more nuanced situations including having players control a positive fraction of the demand or removing the restriction that players choose a single route. After several extensions to more general settings, Roughgarden (2015a) showed that bounds on the PoA also extend to other solution concepts such as the mixed Nash equilibrium, the correlated equilibrium and the coarse correlated equilibrium.

More recently, there has been ample interest in understanding the stochastic aspects of the game, both on the supply and demand sides. The interest comes from the need to understand how the uncertainty of latencies—manifested through bad weather or accidents in road traffic applications, or jitter and failures in telecommunication applications—and the uncertainty of traffic patterns—what origins, destinations and amount of demand materialize in practice—affect the strategies and behaviour of agents playing the game. In this work, we go back to the selfish routing model with atomic players put forward by Rosenthal (1973). Inspired by that, we study congestion games but adding uncertainty to the demand side. Suppose one is planning a trip from home to work for the tomorrow morning commute. From routinely making that trip, one probably knows the possible routes and one might also have an idea of the potential number of drivers that may be on the road during an average morning. However, how many actual drivers will be there? It is likely that there is some deviation from the expected number, and that variability may imply that it is best for one to anticipate and plan the best route accordingly. As it is common in the literature, we assume that route planning happens before getting any signal of the realized congestion in the morning. This situation can be modeled as a game of incomplete information.

For simplicity, we shall assume that each driver will be present independently at random with a certain probability , which we consider homogeneous across players. With probability

, the player will stay at home and not take the trip. Concretely, the number of participating players in the game is represented by a binomial random variable with probability

and total number of players . We assume that and are common knowledge among all players, but the realization of the random variable is unknown to the players.

The model boils down to an atomic congestion game in which there is uncertainty about the number of players involved. The question we pose is how players should strategize in this setting. In answering this question, we explore the impact caused by both the individual optimization of routes and the uncertainty in who shows up. Even though the game we consider is a game of incomplete information, we show how the game can be transformed into a deterministic one by appropriately adapting the cost functions. This allows us to analyze it as a (regular) complete information game. For these transformed cost functions, we derive tight bounds for the PoA by using the -smoothness framework first explored by Harks and Végh Harks and Végh (2007) and thoroughly studied by Roughgarden (2015a).

### Our contribution

There are two key features of our analysis that make it different from previous studies with similar motivations. First, we consider that the social-planner solution—the yardstick that all the PoA papers have used to evaluate the efficiency of equilibria—is exposed to the same uncertainty. The routes that are assigned to players have to be set before knowing whether the player is actually present or not. For the players that show up, the corresponding routes are used; for the others, the routes are discarded. This sets the bar for a fairer comparison since in both situations we have similar uncertainty. Second, Roughgarden (2015b) showed that PoA bounds for smooth games also apply to Bayesian extensions of the game as long as types are drawn independently. His goal is to find bounds that are independent of the distribution over types, which means that taking a worst-case approach in terms of the Bayesian game reduces to the case of , and does not provide additional information for the case of stochastic players. Instead, we study a weaker definition of smoothness and we parameterize our bounds explicitly with respect to . When is close to zero one would expect that the PoA is close to 1, whereas when is close to one one would expect that the PoA reduces to the deterministic case. Through our model, we can show how the actual curve behaves between those extremes and whether the intuition above materializes.

We assume that cost functions are affine and obtain the following results. In Theorem 4.1, we establish (tight) upper bounds for the PoA. We show that the PoA is nondecreasing in and has two kinks. In fact, we show that the range of can be divided into three regions: , and , where and is the real root of . In the lower range, the PoA is at most , which is equal to the PoA for nonatomic congestion games with affine costs. In the middle range, the PoA is at most

 1+p+√p(2+p)1−p+√p(2+p),

and in the upper range, the PoA is at most

 1+p+p21+p.

For we get back the bound known for deterministic atomic congestion games, whereas for all we have a smaller bound. Note the (perhaps unexpected) fact that for we get the significantly smaller and constant bound of , and that this holds independently of the structure of the congestion game and for any number of players.

We show that all the bounds mentioned above are tight. In fact, these bounds are attained (asymptotically) by means of standard routing congestion games and with purely linear costs. Example 4.1 shows that the bound is asymptotically tight for in the lower range by considering a sequence of bypass networks. Notice that even though cost functions are linear, due to the randomness in demand, the adjusted cost functions have an intercept term implying that the PoA equals to , which is the PoA for nonatomic congestion games with affine costs, and not as one could expect from nonatomic congestion games with purely linear costs. Example 4.2 shows that the PoA is asymptotically tight for values of in the middle range by considering a sequence of roundabout networks. Finally, Example 4.3 shows that the PoA is tight for values of in the upper range by considering a simple variant of the game defined by Awerbuch et al. (2013).

## 2. Related literature

The idea of systematically measuring the inefficiency of equilibria started with Koutsoupias and Papadimitriou (1999, 2009); Papadimitriou (2001) who defined the PoA as the ratio of the social cost of the worst equilibrium over the optimum social cost. The idea has been extensively applied to various classes of games, in particular to congestion games in general and to routing games in particular.

Bounds for the PoA are substantially different for atomic and nonatomic congestion games. In nonatomic games the equilibrium concept is due to Wardrop (1952), and its properties have been thoroughly studied, among others, by Beckmann et al. (1956). Bounds for the PoA in these games can be found in Roughgarden and Tardos (2002, 2004); Roughgarden (2003, 2005) and Correa et al. (2004, 2008). These papers provide tight bounds for the PoA when the cost functions are restricted to specific classes—such as polynomials—and show that the bounds depend only on the cost functions but not on the topology of the network. We refer the reader to Roughgarden (2007); Roughgarden and Tardos (2007); Correa and Stier-Moses (2011) for surveys of these results. On the other hand, atomic congestion games, and in particular atomic routing games, were studied by Rosenthal (1973). The PoA for these games is examined in Awerbuch et al. (2013), Christodoulou and Koutsoupias (2005) and Suri et al. (2007) for the case of weighted and unweighted atomic players. Aland et al. (2011) provided exact bounds for the PoA when costs are polynomial functions. As an extension of the previously cited results, Roughgarden (2015a) studied a property, called -smoothness, that allows to easily find bounds for the PoA. The bounds obtained by this technique hold not only for pure equilibria, but also for mixed, correlated, and coarse-correlated equilibria. Our results are obtained through an adaptation of the -smoothness idea.

Several papers have examined the behavior of the PoA as a function of some parameter of the model. Correa et al. (2008) provided an expresion to compute the PoA as a function of the maximum congestion level in the network. Relatedly, Colini-Baldeschi et al. (2019, 2017) studied the asymptotic behavior of the PoA in light and heavy traffic regimes, and showed that under weak conditions efficiency is achieved in both limit cases. Similar results for congestion games in heavy traffic have been obtained in Wu et al. (2017, 2018), using a slightly different technique.

Lately, there has been an increased interest in understanding the stochastic aspects of the game, both on the supply and demand sides by, for instance, introducing some risk posture in the model (see, e.g., Ordóñez and Stier-Moses, 2010; Nikolova and Stier-Moses, 2014; Lianeas et al., 2019). Also, while most results concern the PoA and its bounds for games with complete information, attention has recently turned to incomplete information games. Gairing et al. (2008) studied the PoA for congestion games on a network with capacitated parallel edges, where players are of different types—the type of each agent being the traffic that she moves—and types are private information. Wang et al. (2014) considered nonatomic routing games with random demand and examine the behavior of the PoA as a function of the demand distribution. Roughgarden (2015b) showed that whenever player types are independent, the PoA bounds for complete information games extend to Bayesian Nash equilibria of the incomplete information game. In particular, for congestion games with affine costs, the bound of holds also for games of incomplete information. The definition of smoothness in that framework is stronger than the definition we use and therefore has stronger implications, as the quality of Bayesian Nash equilibria is compared to an adaptive definition of optimum. However, because Roughgarden’s bounds are more robust, they are not as sharp as ours. Recently, Correa et al. (2019) proved a similar result for smooth games with stochastic demand and arbitrary correlations in playing probability, but their bounds are not tight for a fixed .

Our model can also be seen as a special case of the perception-parametrized affine congestion games studied by Kleer and Schäfer (2019). These games feature two parameters and which affect respectively the cost perceived by the players and the cost considered by the central planner. Our model corresponds to the case where . Focusing on the results which apply to this case, we observe that their Theorem 2 establishes the same bound for all . Our results improve both the upper and lower bounds for . Indeed, on the one hand we show that their upper bound is valid and tight on the larger interval , and on the other hand we also obtain tight bounds outside this interval. Kleer and Schäfer (2019) also investigated the price of stability (PoS). Their Theorem 5 characterizes the PoS for all showing that . Naturally, this upper bound on is strictly smaller than our two upper bounds for in the corresponding range . On the other hand, for our Example 4.1 has a unique equilibrium and thus gives a larger lower bound of , which combined with our upper bound on shows that in fact this is sharp with in this range. Our result thus complements theirs and completes the characterization of the PoS.

Let us finally mention some other papers that investigate the PoA in stochastic models. For instance, Stidham (2014) studied the efficiency of some classical queueing models on various networks, whereas Hassin et al. (2018) examined a queueing model with heterogeneous agents and studied how the PoA varies with the intensity function.

## 3. Congestion games with stochastic demand

### 3.1. Atomic congestion games

Consider a finite set of players and a finite set of resources . The set of feasible strategies for player is and we denote . For each and each , call

 ne(s)=|{j∈N∣e∈sj}|

the number of players who use resource when the strategy profile is . Let be a weakly increasing function such that is the cost experienced by each player who uses resource and there are of them. Given a strategy profile the cost of player is given by

 Ci(s)=∑e∈sice(ne(s)). (3.1)

The above quantities define an atomic congestion game (ACG) . This class of game was introduced in Rosenthal (1973).

The strategy profile is a pure Nash equilibrium (PNE) if for all and we have

 Ci(^s)≤Ci(si,^s−i).

We call the set of pure Nash equilibriums of . The set is non-empty since is a potential game (see Rosenthal, 1973; Monderer and Shapley, 1996).

Consider now the social cost (SC)

 C(s)=∑i∈NCi(s)=∑e∈Ene(s)ce(ne(s)). (3.2)

A strategy profile minimizing this aggregate cost is called a social optimum (SO) and is called the socially optimum cost (SOC).

Given an atomic congestion game (ACG) , the price of anarchy (PoA) and price of stability (PoS) are defined as

where is a social optimum (SO).

A very useful tool to bound the PoA in a class of games is the concept of -smoothness.

###### Definition 3.1.

A game is said to be -smooth if, for all strategy profiles , we have

 ∑i∈NCi(s′i,s−i)≤λ⋅C(s′)+μ⋅C(s). (3.3)
###### Lemma 3.2 (Roughgarden (2015a)).

If the game is -smooth with and , then

 PoA(Γ)≤λ1−μ. (3.4)

### 3.2. Stochastic demand

We consider an atomic congestion game with stochastic demand (ACGSD) in which each player participates in the game with probability , and is inactive—hence incurring no costs—with probability . As most prior work, each player chooses her strategy before knowing Nature’s draw.111Since players could update their priors after starting the trip and switch to other routes, (Nguyen and Pallottino, 1988; Miller-Hooks, 2001; Marcotte et al., 2004) consider a richer set of strategies called hyperpaths. The random variable indicates whether player is active. The variables are assumed to be independent. We denote the resulting atomic congestion game with stochastic demand (ACGSD).

In a Bayesian Nash equilibrium (BNE), each player minimizes her expected cost with respect to her beliefs about the presence of other players. In the current setting, the expected cost for a player using resource is given by

 cpe(x)=p⋅E(ce(1+X)), (3.5)

where and is the number of players—active and inactive—who choose to use that resource. Hence, for a given strategy profile the total expected cost for a player is now

 Cpi(s)=∑e∈sicpe(ne(s)). (3.6)

and is a Bayesian Nash equilibrium (BNE) if for all and we have

 Cpi(^s)≤Cpi(si,^s−i).

It follows that a ACGSD is equivalent to a standard ACG in which the costs are replaced by . This immediately implies that a BNE always exists.

The social expected cost (SEC) is now

 Cp(s)=∑i∈NCpi(s)=∑e∈Ene(s)cpe(ne(s))=E(∑e∈ENe(s)⋅ce(Ne(s))), (3.7)

where and the last equality follows from the law of total expectation, by writing and conditioning on .

Call the class of all ACGSDs having affine costs

 ce(x)=ae⋅x+be,with ae,be≥0,∀e∈E. (3.8)

We are interested in the best possible bounds for the PoA and PoS over this class. To this end, we define

## 4. Price of Anarchy for congestion games with stochastic demand and affine costs

For the rest of this paper we focus on atomic ACGSDs having affine costs. We will prove -smoothness bounds that yield tight bounds for the PoA over the class .

### 4.1. Upper bounds

The following is the main result of our work. It provides an upper bound on the PoA for ACGSDs with affine costs. Define and as follows:

 ¯p0=1/4,¯p1∼0.3774 is the real root of 8p3+4p2=1. (4.1)
###### Theorem 4.1.

Let be defined as in Eq. 4.1. Then every ACGSD is -smooth, with

 (λ,μ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,14) if  0

hence,

 PoA(p)≤⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩43 if  0

Fig. 1 shows the graph of the bounds in Eq. 4.3.

The idea of the proof of Theorem 4.1 is the following. First we show that, under some conditions, for fixed , all games are -smooth. Then we optimize over the parameters and and we show that the the optimal take different values in three regions of the interval . Using then Lemma 4.2, we find upper bounds for the PoA. The following lemma is a slight variation of the technical lemma in Christodoulou and Koutsoupias (2005); Aland et al. (2011) that takes into account the role of the probability .

###### Lemma 4.2.

Let and be such that

 k(1+mp)≤λk(1−p+pk)+μm(1−p+pm)∀k,m∈N. (4.4)

Then every ACGSD is -smooth.

###### Proof.

We first observe that Eq. 4.4 implies that (by considering and ), (by considering and ), and for all

 kcpe(m+1) =kp[ae(1+mp)+be] ≤p[λkae(1−p+pk)+μmae(1−p+pm)+kbe] (4.5) ≤λkcpe(k)+μmcpe(m),

where the first inequality follows from Eq. 4.4. For all

 ∑i∈N∑e∈s′icpe(ne(s′i,s−i)) ≤∑i∈N∑e∈s′icpe(ne(s)+1) =∑e∈Ene(s′)⋅cpe(ne(s)+1) ≤∑e∈Eλ⋅ne(s′)⋅cpe(ne(s′))+μ⋅ne(s)⋅cpe(ne(s)) =λ⋅Cp(s′)+μ⋅Cp(s),

where the second inequality follows from Eq. 4.5 with and . ∎

The optimization of Eq. 4.4 over and is similar to what has been done by Aland et al. (2011) and Roughgarden (2015a) for deterministic congestion games. For the condition in Eq. 4.4 is trivially satisfied as long as , so it suffices to consider . Hereafter we let and we denote the set of all pairs with . Then the smallest possible is

 λ =sup(k,m)∈Pk(1+pm)−μm(q+pm)k(q+pk) =sup(k,m)∈Pμ[k(1+pm)−m(q+pm)k(q+pk)]+(1−μ)1+pmq+pk

and the best bound on the PoA that we can get is

 PoA(p)≤B(p)\coloneqqinfλ,μλ1−μ=infμ∈[0,1)sup(k,m)∈Pμ1−μ[k(1+pm)−m(q+pm)k(q+pk)]+1+pmq+pk.

Denoting

 y=μ1−μ∈[0,∞) (4.6)

and introducing the functions

 φk,mp(y) =y[k(1+pm)−m(q+pm)k(q+pk)]+1+pmq+pk, (4.7) φp(y) =sup(k,m)∈Pφk,mp(y), (4.8)

the optimal bound is

 B(p)=infy≥0φp(y). (4.9)

If this infimum is attained at a certain then we get together with the corresponding optimal parameters

 μ=yp1+ypandλ=(1−μ)B(p)=φp(yp)1+yp.

Our next lemma gathers some basic facts about the function and shows in particular that its infimum is attained.

###### Lemma 4.3.

The function has the following properties:

1. is convex.

2. and for all .

3. for .

4. The minimum of is attained at a point .

###### Proof.
• This property is obvious since is a supremum of affine functions.

• For we may take and to get . Now, consider the case and rewrite the expression of as

 φp(y)=supk≥11k(q+pk)[(y+1)k+supm≥0((y+1)pk−yq)m−ypm2]. (4.10)

Relaxing the inner supremum and considering the maximum with we get

 φp(y)≤supk≥11k(q+pk)[(y+1)k+((y+1)pk−yq)24yp].

The expression within square brackets is a quadratic so the conclusion follows by noting that all three expressions

 k2k(q+pk), kk(q+pk), 1k(q+pk)

are bounded for .

• This follows directly from .

• This is a consequence of (1), (2), (3). ∎

We now show that when is sufficiently large the minimum of is attained at a point for which the supremum in Eq. 4.8 is reached with and simultaneously for and , that is,

 φp(yp)=φ1,1p(yp)=φ1,2p(yp).

For smaller values of the supremum in Eq. 4.8 for is still reached at with either or , but also for and tending to . In order to state this more precisely we introduce the limit function

 φ∞(y)=limk→∞supm≥0φk,mp(y).

Our next lemma shows that this limit is well defined and does not depend on . Note that this function is also a minorant of .

###### Lemma 4.4.

For all we have

 φ∞(y)=(y+1)24y.

Moreover,

 φ′∞(y)=y2−14y2<0

for all .

###### Proof.

Fix and . The maximum of for is attained at the integer that is closest to the unconstrained (real) maximizer

 ˆm=(y+1)pk−yq2py. (4.11)

For large we have and we may find such that . Then,

 supm∈N((y+1)pk−yq)m−ypm2 =yp(2ˆm−¯¯¯¯¯m)¯¯¯¯¯m =yp(ˆm+f)(ˆm−f) =yp(ˆm2−f2) =((y+1)pk−yq)24yp−ypf2,

from which it follows that

 φ∞(y)=limk→∞(y+1)k+14yp((y+1)pk−yq)2−ypf2k(q+pk)=(y+1)24y.\qed

In the sequel we will find the exact expression for the optimal solution for all . To this end, consider the solutions of the equations

 φ1,0p(y)=φ∞(y) ⟺y=y0,p\coloneqq13 φ1,1p(y)=φ∞(y) ⟺y=y1,p\coloneqq11+2p+2√p(2+p) φ1,1p(y)=φ1,2p(y) ⟺y=y2,p\coloneqqp1+p.

Note that these three solutions belong to . Let also be the point at which , and the unique real root of which is the point where .

###### Proposition 4.5.

The minimum of is attained at if and only if .

###### Proof.

We claim that

 φp(y0,p)=φ1,0p(y0,p)=φ∞(y0,p) (4.12)

iff .

Since both and are minorants of , their slopes and are subgradients of at . Hence and is indeed a minimizer.

We now establish the claim in Eq. 4.12, that is,

 y0,p[k(1+pm)−m(1−p+pm)k(1−p+pk)]+1+pm1−p+pk≤(1+y0,p)24y0,p∀(k,m)∈P (4.13)

iff .

Substituting , this can be written equivalently as

 0≤p(2k−m−1)2+m(1−3p)−p∀(k,m)∈P.

This holds trivially for so we just consider . Now, for this requires . Conversely, if we have and therefore increases with so that

 p(2k−m−1)2+m(1−3p)−p≥m(1−3p)−p≥1−4p≥0.\qed
###### Proposition 4.6.

The minimum of is attained at if and only if .

###### Proof.

We claim that iff we have

 φp(y1,p)=φ1,1p(y1,p)=φ∞(y1,p), (4.14)

so that and are subgradients of at . Now, since , by Lemma 4.4, we have so that and therefore is a minimizer.

We prove Eq. 4.14, that is,

 y1,p[k(1+pm)−m(1−p+pm)k(1−p+pk)]+1+pm1−p+pk≤(1+y1,p)24y1,p∀(k,m)∈P, (4.15)

iff .

Dividing by and letting

 z=1+y1,p2y1,p=1+p+√p(2+p),

this becomes

 [k(1+pm)−m(1−p+pm)k(1−p+pk)]+1+pm1−p+pk(2z−1)≤z2.

Multiplying by and factorizing, this can be rewritten as

 0≤p(zk−m+(1−p)z−22p)2+((1−p)z−1−p)m−((1−p)z−2)24p, (4.16)

so that, denoting this last expression, Eq. 4.16 is equivalent to for all . We observe that

 Qp(1,0)≥0 ⟺ z≥2 ⟺ p≥¯p0 (4.17) Qp(1,2)≥0 ⟺ 8p3+4p2−1≤0 ⟺ p≤¯p1 (4.18)

so that is a necessary condition for Eq. 4.15. We now show that it is also sufficient.

###### Case 1.

: The inequality is equivalent to so that the most stringent condition is for , which holds for all as already noted in Eq. 4.17.

###### Case 2.

: From the very definition of we have that Eq. 4.15 holds with equality for , so that . Since is quadratic in , in order to have for all , it suffices to check that . The latter can be factored as

 Qp(2,1)=2(1+p)z(z−2)+1,

so that, substituting and simplifying, the resulting inequality becomes

 4p(1+p)√p(2+p)+4p3+8p2+2p−1≥0.

The conclusion follows since this expression increases with and the inequality holds for .

###### Case 3.

: As noted in Eq. 4.17 we have for all . On the other hand, since we have that increases for , so that it suffices to show that . Now, can be factored as

 Qp(2,2)=2(1+p)(z−1)2−4pz

and substituting we get

 Qp(2,2)=4p2(1+p+√p(2+p))≥0.
###### Case 4.

: Let be the slope of the linear term in . Neglecting the quadratic part we have

 Qp(k,m)≥αm−((1−p)z−2)24p (4.19)

and therefore it suffices to show that the latter linear expression is non-negative. We claim that for all we have . Indeed, substituting we get

 α=(1−p)√p(2+p)−p(1+p),

so that if and only if which simplifies as and holds for , and in particular for . Thus the right hand side in Eq. 4.19 increases with , so it remains to show that it is non-negative for . The latter amounts to

 3α≥((1−p)z−2)24p,

which is equivalent to

 2(6p+1+p2)(1−p)√p(2+p)≥1+2p+11p2+12p3+2p4

and can be seen to hold for all . ∎

###### Proposition 4.7.

The minimum of is attained at if and only if .

###### Proof.

For and the unconstrained maximizer Eq. 4.11 is so that is attained at and . The slopes of the corresponding terms are

 (φ1,mp)′(y)={pif m=1,−1if m=2.

If the outer supremum is attained for it follows that and therefore is a minimizer. Considering the expression Eq. 4.10 and substituting the value of , it follows that is attained at if and only if

 (1+2p)k+supm∈N((1+2p)k−q)pm−p2m2≤[(1+p)2+p2]k(q+pk)∀k≥2. (4.20)

We claim that this holds if and only if . To this end we note that for all the unconstrained maximum of the quadratic is attained at

 ˆm=(1+2p)k−q2p>1.

Proceeding as in the proof of Lemma 4.4 we may find an integer and such that , hence, the supremum for is attained at and

 supm∈N((1+2p)k−q)pm−p2m2=p2(ˆm2−f2)=14((1+2p)k−q)2−p2f2. (4.21)

Replacing this expression into Eq. 4.20 and after simplification the condition becomes

 0≤[8p3+4p2−1]k2+[2−2p−4p2−8p3]k+4p2f2−(1−p)2∀k≥2. (4.22)

It follows that a necessary condition is which amounts to . It remains to show that once the inequality Eq. 4.22 holds automatically. Consider first the case . Ignore the non-negative term and denote

 Q(x)=[8p3+4p2−1]x2+[2−2p−4p2−8p3]x−(1−p)2.

For this is quadratic and convex in and we have

 Q′(3)=40p3+20p2−2p−4≥0∀p≥¯p1.

Hence is increasing for and then Eq. 4.22 holds for all since

 Q(k)≥Q(3)=48p3+23p2−4p−4≥0∀p≥¯p1.