lecture 01 - iran university of science and...

Lecture 01Efficiency, Parteo-Optimality, and

Fairness

Introduction

• In the context of resource allocation, welfare analysis uses aneconomic approach to study the overall benefit (or welfare)generated under alternative mechanisms for allocation of scarceresources.

• For our purposes, welfare analysis serves as a benchmark approachto resource allocation in engineered systems, and in particularallows us to introduce several essential concepts in economicmodeling.

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Lecture Overview

• We begin by introducing the notion of utility, the value that is derived by anindividual from consumption of resources.

• Next, we discuss efficiency and define Pareto optimality, a way tomeasure the welfare of allocation choices.

• We discuss fairness considerations, and in particular how different notionsof fairness lead to different ways of choosing between Pareto optimaloutcomes.

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Utility

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Utility• Utility provides a basic means of representing an individual’s preferences

for consumption of different allocations of goods or services.

• We start by defining a preference relation for an agent:

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Utility• “Utility” refers to an assignment of value to each possible bundle that is

aligned with an agent’s preferences. Formally, a preference relation ≽ isrepresented by a utility function

• In some examples, the user’s preferences and so their utility may dependonly on a lower-dimensional function of the resource bundle;

for example, in the case of resource allocation in networks, a user’sutility may be a function only of the total rate allocation they receive.

In these cases we adapt our notation accordingly.

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Assumptions on Utility Functions

Assumption 1. Monotonicity:

We only consider utility functions that are non-decreasing, i.e.,if every component of the allocated resource vector weaklyincreases, then the utility weakly increases as well.

This means that every individual prefers more resources to lessresources.

A key implicit reason that monotonicity is plausible is the notion of freedisposal: that excess resources can always be “disposed of” withoutany cost or penalty. With free disposal, an individual is only ever betteroff with additional resources.

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Assumptions on Utility Functions

Assumption 2. Concavity:

• Utility functions can take any shape or form, but one important distinction is between concave and convex utility functions.

Concave utility functions represent diminishing marginal returns to increasing amounts of goods,

Convex utility functions represent increasing marginal returns to increasing amounts of goods.

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A Wireless Communications Example

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Another Interpretation of Concavity

• In the context of network resource allocation, there is anadditional interpretation of concavity that is sometimes useful.

Consider two users with the same utility function U(x) as afunction of a scalar data rate allocation x;These users share a single link of unit capacity.We consider two different resource allocations. In the first, we randomly allocate the entire link to one of the

two users. In the second, we split the link capacity equally between the

two users.Which scenario do the users prefer?

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Concavity: discussion• Observe that if the users are making a phone call which requires the entire

link for it’s minimum bit rate, then:

The second scenario is undesirable!

• On the other hand, if the users are each downloading a file, then:

The second scenario delivers a perfectly acceptable average rate (giventhat both users are contending for the link).

• The first case is often modeled by assuming that U is convex because theexpected utility to each user in the first allocation is higher than theexpected utility to each user in the second allocation.

(This result follows from: Jensen’s inequality.)

• Concave utility functions are sometimes thought to correspond to elastictraffic and applications, such as “file sharing” downloads;

• Convex utility functions are often used to model applications with inelasticminimum data rate requirements, such as voice calls. 12

Example utility functions for elastic and inelastic traffic

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Efficiency &

Pareto-Optimality

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Efficiency and Pareto Optimality• A central question in any resource allocation problem is to define

the notion of efficiency.

To an engineer, efficiency typically means that all resources arefully utilized.

However, this leaves many possibilities open; in particular, howare those resources allocated among agents in the system?

It may be possible that some allocations that fully utilize resourceslead to higher overall utility than others.

In economics, efficiency is typically tied to the utility generated byresource allocations, through the notion of Pareto optimality.

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Pareto Dominance & Pareto Optimality• Let R be the set of users in a system, and let X denote the set of all

possible resource allocations.

• In other words, the allocation x leaves everyone at least as well off (in utility terms), andat least one user strictly better off, than the allocation y.

• From a system-wide (or “social”) standpoint, allocations that are Pareto dominated areundesirable, since improvements can be made to some users without affecting others.

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Economic efficiency vs. engineering efficiency• In the single resource setting, an economist’s notion of efficiency

corresponds with an engineer’s notion of efficiency: the resourceshould be fully utilized.

• However, in general, economic efficiency is not only concerned withfull utilization of available resources, but also optimal allocation ofthose resources among competing users.

• This is the additional twist in the second example:

it is essential to consider the value generated in consideringwhich user receives each resource.

Indeed, that challenge is at the heart of the economic approachto resource allocation.

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Fairness

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Fairness

• As the examples in the previous section illustrate, there may be many Pareto optimal allocations. How do we choose among them? To answer this, we will focus on fairness considerations in

resource allocation. A central point is to distinguish between “Pareto optimality”,

which is an objective notion, and “fairness”, which is a subjective notion.

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Fairness

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Fairness

• Observe that each choice of f defines a potentially different choice of

the Pareto optimal point x∗. (Indeed, it can be shown that under our

assumptions, every Pareto optimal point can be recovered through an

appropriate choice of f.)

• We interpret f itself as encoding a fairness criterion in this way: the

choice of f directly dictates the resulting “fair” allocation among allPareto optimal points. 24

Fairness

• We note that, of course, there are many generalizations of thisapproach to fairness.

For example, the function f may be user-dependent, or there may be aweight associated to each user in the objective function above.

More generally, the aggregation function in the objective need not be asummation;

Next, we introduce three fairness criteria identified in this way

(utilitarian, proportional, and α-fairness), and a fourth obtained

as a limiting case (max-min fairness).25

Utilitarian Fairness

• Perhaps the simplest choice of fairness is the case where f is theidentity, in which case the Pareto optimal point that maximizesthe total utility of the system is chosen.

This is called the utilitarian notion of fairness;

An alternative interpretation is that utilitarian fairnessimplicitly assumes that all agents’ utilities are measured inexactly the same units; and given this fact, the allocationshould be chosen that maximizes the total “utility units”generated.

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Proportional Fairness

• Utilitarian fairness can lead to allocations that remove users withlower utilities from consideration, in favor of those that generatehigher utilities.

• One way of mitigating this effect is to scale down utility at higherresource allocations.

• This leads to the notion of proportional fairness, obtained byusing f(U) = logU.

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Proportional Fairness

• This property states that, under any other allocation, the sum of proportional changes

in the users’ utilities will be non-positive.

• Thus, if some User A’s share increases, then there will be at least one other user

whose share will decrease and further,

the proportion by which it decreases will be larger than the proportion by which the

share increases for User A. Therefore, such an allocation is called proportionally

fair.

• If the fairness criterion is chosen such that f(Ur) = wrlogUr, where wr is some weight,then the resulting allocation is said to be weighted proportionally fair. 28

• Then if x∗ is the resulting optimal allocation, and x is any other allocation, thefollowing inequality holds (see the next slide for a proof):

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α-fairness• Consider the following fairness criterion:

• This fairness function has the property that it is strictly concave and strictly increasing for all α ≥ 0.

• As α increases, the fairness function exhibits progressively stronger decreasing marginal returns.

• The resulting family of fairness notions, parameterized by α, is called α-fairness.

• This family is particularly useful because it includes several special cases as aresult.

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α-fairness

• when α = 0, we recover utilitarian fairness;

• and when α = 1, we recover proportional fairness.

• The case α = 2 is sometimes called TCP fairness in the literature on congestioncontrol in communication networks, because the allocation it leads to mimicsthe allocations obtained under the TCP congestion control protocol.

• Finally, an important special case is obtained when α→∞; we discuss this next.31

Max-min fairness

• A max-min fair allocation satisfies the following property:

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Max-min fairness

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Contradicts with 𝒙 beingmax-min fair!

Proof by contradiction:

Max-min fairness

•X^ is max-min fair if it solves the following optimization problem:

• It can be shown that this outcome is obtained as the limit of the α-fairallocation, as α→∞.

• Max-min fairness is sometimes called Rawlsian fairness after thephilosopher John Rawls.

• It advocates for the protection of the utility of the least well off user in thesystem, regardless of whether a small utility change to this user might causelarge changes in the utility of another user.

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Max-min fairness

• This formal definition corresponds to the following operational definition:

• Consider a set of sources 1, ..., n that have resource demands x1, x2, ..., xn.

• Without loss of generality, order the source demands so that x1 <= x2 <= ... <= xn.

• Let the server have capacity C.

• Then, we initially give C/n of the resource to the source with the smallest demand, x1.

This may be more than what source 1 wants, perhaps, so we can continue the process.

• The process ends when each source gets no more than what it asks for, and, if its demand was not satisfied, no less than what any other source with a higher index got.

• We call such an allocation a max-min fair allocation, because it maximizes the minimum share of a source whose demand is not fully satisfied.

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Fair Share of a Resource

desired: 1/8

desired: 1/3

desired:

2/3

P3

P2

P1

Max-Min Fair Share (1)

desired: 1/8

Fair share: 1/3 each

1. Satisfy customers who need less than their fair share

2. Split the remainder equally among the remaining customers

Return surplus:

1/3 1/8 = 5/24New fair share

for P2 & P3:

1/3 + ½ (5/24) each

P1

P3

P2

Max-Min Fair Share (2)

received: 1/8

Fair share:

1/3 + ½ (5/24) each

1. Satisfy customers who need less than their fair share

2. Split the remainder equally among the remaining customers

Return surplus:

1/3 + ½ (5/24) 1/3

= ½ (5/24)

Remainder of

1/3 + 2 ½ (5/24)

goes to P2

P1

P3

P2

Max-Min Fair Share (3)

received: 1/8

Final fair distribution:

received: 1/3

P1

P3

P2

received: 1/3 + 5/24

deficit: 1/8

Max-Min Fair Share

desired: 1/8

desired: 1/3

desired:

2/3

P3

P2

P1

desired: 1/8

Fair share: 1/3 each

1. Satisfy customers who need less than their fair share

2. Split the remainder equally among the remaining customers

Return surplus:

1/3 1/8 = 5/24New fair share

for P2 & P3:

1/3 + ½ (5/24) each

P1

P3

P2

received: 1/8

Final fair distribution:

received: 1/3

P1

P3

P2

received: 1/3 + 5/24

deficit: 1/8

received: 1/8

Fair share:

1/3 + ½ (5/24) each

1. Satisfy customers who need less than their fair share

2. Split the remainder equally among the remaining customers

Return surplus:

1/3 + ½ (5/24) 1/3

= ½ (5/24)

Remainder of

1/3 + 2 ½ (5/24)

goes to P2

P1

P3

P2

ba

c d

Example: Max-Min Fair Share

Link capacity

= 1 Mbps

Wi-Fi transmitter

(Server)

Application 1

Application 2

Application 3

Application 4

8 packets per sec

L1 = 2048 bytes

40 pkts/s

L4 = 1 KB

25 pkts/s

L2 = 2 KB

50 pkts/s

L3 = 512 bytes

Link capacity

= 1 Mbps

Wi-Fi transmitter

(Server)

Application 1

Application 2

Application 3

Application 4

8 packets per sec

L1 = 2048 bytes

40 pkts/s

L4 = 1 KB

25 pkts/s

L2 = 2 KB

50 pkts/s

L3 = 512 bytes

50 pkts/s

L3 = 512 bytes

Example: Max-Min Fair Share8 2048 + 25 2048 + 50 512 + 40 1024 = 134,144 bytes/sec = 1,073,152 bits/secAppl. A Appl. B Appl. C Appl. D Total demand

available capacity of the link is C = 1 Mbps = 1,000,000 bits/sec

SourcesDemands

[bps]Balances after

1st roundAllocation #2

[bps]Balances after

2nd roundAllocation #3 (Final) [bps]

Final balances

Application 1 131,072 bps 118,928 bps 131,072 0 131,072 bps 0

Application 2 409,600 bps 159,600 bps 332,064 77,536 bps 336,448 bps 73,152 bps

Application 3 204,800 bps 45,200 bps 204,800 0 204,800 bps 0

Application 4 327,680 bps 77,680 bps 332,064 4,384 bps 327,680 bps 0

Weighted Max-Min Fair Share:Source weights : w1 = 0.5, w2 = 2, w3 = 1.75, and w4 = 0.75

Src DemandsAllocation #1 [bps]

Balances after 1st round

Allocation #2 [bps]

Balances after 2nd round

Allocation #3 (Final) [bps]

Final balances

1 131,072 bps 100,000 31,072 122,338 8,734 bps 131,072 bps 0

2 409,600 bps 400,000 9,600 489,354 79,754 bps 409,600 bps 0

3 204,800 bps 350,000 145,200 204,800 0 204,800 bps 0

4 327,680 bps 150,000 177,680 183,508 144,172 bps 254,528 bps 73,152 bps

Max-min fairness

⟹

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Example• We consider a communication network resource allocation problem with three

users and two resources, each of unit capacity.

• Letting xr be the rate allocation to user r, the two resources define two constraints on x:

• Suppose the utility functions of all users are the identity utility function:

Ur(xr) = xr for all r.44

• Thus the utilitarian allocation yields nothing to user 3, instead rewarding theshorter routes 1 and 2.

• The max-min fair allocation, by contrast, allocates the same data rate touser 3 as to users 1 and 2, despite the fact that user 3 uses twice as manyresources.

• Proportional fairness falls in the middle: user 3 receives some data rate, buthalf as much as each of the other two users. In this sense, proportionalfairness “interpolates” between the other two fairness notions; the same istrue for other values of α with 0 < α <∞. 45

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• Up until now, we defined the notions of fairness by simply assuming that the utilityfunction of the users are identity functions of their share; i.e.,

Ur(xr) = xr.

• In fact, when utility functions are a “design choice”, we can directly apply the fairnesscriteria to the user’s share, and interpret the resulting functions as “fair” utility functions.

For example, an𝜶–fair utility function has the following form:

• However, for the cases where utility functions are forced by the system model, we need toapply the fairness criteria to the utility functions themselves, giving rise to the so-called

utility-fairness criteria.

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Utility-Fairness vs. Share-Fairness

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• Consider a network consisting of a single link of capacity one shared by two users.

One user transfers data according to an elastic application with strictly increasing

and concave bandwidth utility U1(·).

The other user transfers real-time video data with a non-concave bandwidth utility

function U2(·).

• If the bandwidth is shared equally, what is referred

to as max-min bandwidth allocation in this

example,

• user 1 receives a much larger utility than user

2.

• Conversely, user 2 would not be satisfied

since he does not receive the minimum video

encoding bandwidth.

• If we want to share utility equally, instead of

bandwidth, we would like to have a resource

allocation, where the received utilities are equal or

utility max-min fair, i.e. U1(x1) = U2(x2) = u∗.

Summary of the 𝜶 −fairness framework

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Subjective vs. objective: fairness vs. efficiency

• Let’s clarify the stark contrast between efficiency (Pareto optimality)and fairness.

Pareto optimality is an objective notion, that characterizes theset of points among which various fairness notions might choose.

Fairness is inherently subjective, because it requires making avalue judgment about which users will be preferred over otherswhen resources are limited.

• Observe that there is no “tradeoff” between fairness and efficiency— the two are complementary concepts.

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Appendix I:Jensen’s Inequality

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A Fact about Uni-Variate Convex FunctionsA differentiable function of one variable is convex on aninterval if and only if the function lies above all of itstangents:

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Jensen’s Inequality• The expected value of the convex function of a random variable

is always ≥ the convex function applied to the expected valueof a random variable .

• If X is a random variable and g is a convex function, then:

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The Intuition behind Jensen’s Inequality

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• Risk aversion. The relationship between convexity and uncertainty is closely connectedto the notion of risk aversion in microeconomic modeling. To see the connection, supposewe let U(w) represent the utility to an individual of w units of currency (or wealth).

• Suppose we offer the individual a choice between W units of wealth with certainty; or agamble that pays zero with probability 1/2, or 2W with probability 1/2.

• Note that the same kind of reasoning as above reveals that:

if U is concave, then the individual prefers the sure thing (W units of wealthwith certainty);

if U is convex, the individual prefers the gamble.

• For this reason we say that:

An individual with concave utility function is risk averse.

An individual with a convex utility function is risk seeking.

• It is generally accepted that individuals tend to be risk averse at higher wealth levels, andrisk neutral (or potentially even risk seeking) at lower wealth levels.

Discussion

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Appendix II:Taylor Series

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• In mathematics, a Taylor series is a representation of a function as an infinite sum ofterms that are calculated from the values of the function's derivatives at a single point.

Taylor series

• The Taylor series of a real or complex-valued function f (x) that is infinitely differentiableat a real or complex number a is the power series:

which can be written in the more compact sigma notation as:

• When a = 0, the series is also called a Maclaurin series.

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