Coordination Mechanisms with Misinformation

Constantinos Varsos

1,2

, Michail Fasoulakis

, Giorgos Flouris

and Marina Bitsaki

Institute of Computer Science, FORTH, Vasilika Vouton, Heraklion, Greece

Department of Computer Science, University of Crete, Heraklion, Greece

Keywords:

Game Theory, Misinformation Games, Mechanism Design, Coordination Mechanisms, Information in

Games, Non-atomic Congestion Games with Parallel Links.

Abstract:

We introduce a novel approach for coordination mechanisms in games, based on the idea of misinforming

players about the game formulation in order to steer them towards a behaviour that leads to an improved

outcome in terms of social welfare. As a use case, we study single-commodity non-atomic congestion games

with parallel links and afﬁne cost functions. We propose a simple mechanism that provides to the players

the right incentives to adopt a socially optimal behaviour by misinforming them with regards to the latency

functions of the links, under various assumptions. We use a metric called the Price of Misinformation to

quantify the effect of misinformation on social welfare (compared to the optimum of the actual game), and

show that our mechanism can minimise this metric, resulting in values that are better than the Price of Anarchy

(i.e., the social outcome without any intervention from the designer).

1 INTRODUCTION

In games with selﬁsh and rational players, the out-

come may be very inefﬁcient in terms of social wel-

fare. The Price of Anarchy, introduced in (Koutsou-

pias and Papadimitriou, 1999), measures how far the

worst Nash equilibrium is from the social optimum.

In this paper, we investigate methods to lead play-

ers’ behaviour to a socially improved outcome. Co-

ordination mechanisms were introduced for this pur-

pose in (Christodoulou et al., 2009; Christodoulou

et al., 2014), in which a theoretical framework is pro-

posed where modiﬁcations of the game lead to a re-

duced fraction, compared to the Price of Anarchy, of

the worst Nash equilibrium in the modiﬁed game to

the social optimum of the original game. This has

been applied to many classes of games such as load

balancing and congestion games (Nisan et al., 2007).

A common assumption in game theory is that

players have complete information concerning the

game of interest, i.e., the set of players, the set of

strategies and the payoffs for each player is com-

mon knowledge among players. In (Varsos et al.,

2021), this assumption is dropped by studying situa-

tions where each player may have a different (and thus

incorrect) perception about the game being played,

while being unaware that she has only a subjective

view of the interaction. To that direction, (Varsos

et al., 2021) agglomerate the objective interaction

(called actual game) and the subjective views, in-

troducing the concept of misinformation games for

normal-form games. A key characteristic of misin-

formation games is that each player’s choices are dic-

tated by her view, whereas her payoff is provided by

the actual game. This leads to the introduction of

a new equilibrium concept, the natural misinformed

equilibrium, which is the set of strategic choices

where no player wants to deviate in her view.

Moreover, a metric, the Price of Misinformation

(PoM), was introduced to measure the impact of mis-

information in games, compared to the socially opti-

mum situation (Varsos et al., 2021). Clearly, misinfor-

mation could lead players to strategic choices that are

different from the ones they would make in the ab-

sence of misinformation. This includes choices that

are actually beneﬁcial (from the perspective of social

welfare) for the players. Inspired by this observation,

we combine misinformation games and coordination

mechanisms to address the following question:

Is it possible for the designer of a game to misin-

form players regarding the game parameters, in order

to provide incentives for a better (or even optimal) be-

haviour in terms of social welfare?

We positively answer this question, and provide a

novel way for applying coordination mechanisms us-

ing the concept of misinformation, thereby establish-

Varsos, C., Fasoulakis, M., Flouris, G. and Bitsaki, M.

Coordination Mechanisms with Misinformation.

DOI: 10.5220/0010836100003116

In Proceedings of the 14th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2022) - Volume 1, pages 237-244

ISBN: 978-989-758-547-0; ISSN: 2184-433X

 2022 by SCITEPRESS – Science and Technology Publications, Lda. All r ights reserved

237

ing a connection between the classical coordination

mechanisms and misinformation games.

As in classical coordination mechanisms, where

the designer modiﬁes the game in order to minimize

the ratio between the worst Nash equilibrium of the

modiﬁed game and the social optimum of the origi-

nal game, we propose a similar approach where the

designer misinforms players. Next, we compare the

worst natural misinformed equilibrium (i.e., the worst

result of misinformation) with the social optimum of

the actual game, that is Price of Misinformation. In-

terestingly, it could be less than the Price of Anarchy,

resulting in an improved behaviour (from the perspec-

tive of social welfare) of the players, compared to the

scenario without misinformation.

A key difference between classical coordination

mechanisms and coordination mechanisms with mis-

information is that in the ﬁrst case the designer can

inﬂuence the actual interaction as a whole, whereas

in the latter the designer determines the subjective

views of the players, i.e., the misinformed views.

Thus, in the ﬁrst case the designer modiﬁes the ac-

tual game speciﬁcation, whereas in the latter the de-

signer changes players’ (subjective) information, but

has no power over the actual game speciﬁcation, In

this paper, we consider the setting where the designer

cannot impose a different game speciﬁcation, but can

misinform players about the actual set up.

We consider the problem under assumptions about

the number of misinformed views that the designer

can spawn. In particular, the designer has bounded ca-

pabilities with regards to the number of different mis-

informed views that can be spawned (see Section 6).

As a real-life situation, consider the case where

two companies compete with each other in a market.

We can see one of them (say D) as the designer, and

the other (say P) as a group of divisions (P

,...,P

representing players. The designer D can use a coor-

dination mechanism with misinformation in order to

beneﬁt from the (in-)efﬁciency of the rival company,

by misleading the divisions of P into actions that are

not optimal for P. Although this is a rather simplistic

scenario, in general, coordination mechanisms with

misinformation can be used in (among others) bar-

gaining, negotiation and conﬂict scenarios.

We apply the above ideas for single-commodity

non-atomic congestion games with n parallel links

and afﬁne cost functions. We ﬁrst adapt the concept

of misinformation to the class of non-atomic conges-

tion games (Section 4), and design a polynomial-time

algorithm for computing a pure Nash equilibrium in a

network (Section 5). Moreover, we describe a mecha-

nism for designing misinformation games with an op-

timal Price of Misinformation (and thus better social

outcomes) under various assumptions (Section 6).

2 RELATED WORK

The idea of designing mechanisms to improve coordi-

nation in multi-player systems with selﬁsh players is

not new. One approach is to introduce taxes ((Fleis-

cher et al., 2004), (Fotakis and Spirakis, 2008), (Cara-

giannis et al., 2010), etc.); this has been applied in

congestion games, where players pay a toll for every

edge they use. In (Cole et al., 2003) it was shown that

there exist taxes that reduce the Price of Anarchy to

1. However, there are two issues for this approach:

i) taxes may be very high, and ii) if taxes are part of

the cost, then the Price of Anarchy is not improved.

Similarly, (Lavi and Swamy, 2007), (Seregina et al.,

2017), (Turrini, 2013) used rewards to improve coor-

dination. In (Monderer and Tennenholtz, 2003) the

game is extended by adding new strategies and en-

hanced monetary policies for the players, such that all

Nash equilibria of the new game involve strategies of

the original game exclusively. In that case, the Price

of Anarchy is decreased.

In this paper, we will use the concept of misinfor-

mation games (Varsos et al., 2021). This concept lies

in the area of games with misperceptions (see Chapter

12 in (Luce and Raiffa, 1957)), which studies games

where players have a subjective knowledge regard-

ing game speciﬁcations. Related ideas include: i) hy-

pergames ((Bennett, 1980; Sasaki and Kijima, 2012;

Bakker et al., 2021) etc.), ii) games with unaware-

ness/awareness ((Copic and Galeotti, 2006; Halpern

and R

ego, 2014; Schipper, 2017; Feinberg, 2020)

etc.), and iii) other works ((Esponda and Pouzo, 2016;

Ozbay, 2007) etc.) where players may have misper-

ception. To the best of our knowledge, there has been

no study in this area regarding non-atomic congestion

games, and also none of these has been applied to im-

prove the social outcome of games.

Another approach dealing with unawareness is

that of incomplete information that is described by

Bayesian games. They were introduced by (Harsanyi,

1967) and have received a lot of attention by many

scholars ((Sz

ekely and Rizzo, 2007; Zamir, 2009;

Myerson, 2004), etc.). The main idea of this approach

is that players’ beliefs with regards to game speciﬁca-

tions are represented by probability distributions.

There are several key differences between misin-

formation and Bayesian games. Firstly, in Bayesian

games, players are aware of the incompleteness of

their information (as represented by the probabil-

ity distributions), whereas in misinformation games

players are unaware of the fact that their game spec-

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

238

iﬁcation is wrong. Therefore, they will not consider

mitigating strategies “just in case” they have a wrong

speciﬁcation. Moreover, the payoffs they get are dic-

tated by the actual game and may be radically differ-

ent from the ones speciﬁed in their own game.

Moreover, in the Bayesian approach the concept

of non-common knowledge is dealt indirectly through

the assignment of distributions over actions and be-

liefs, as opposed to misinformation games, where

we can directly deal with non-common knowledge

games, using subjective views. Consequently, the two

aforementioned approaches have different equilibria

concepts, namely Bayesian Nash equilibrium and nat-

ural misinformed equilibrium, respectively.

Another approach to cope with incomplete infor-

mation in non-atomic routing games was introduced

in the works of (Meir and Parkes, 2015a; Meir and

Parkes, 2015b; Meir and Parkes, 2018), where the au-

thors propose a model with player-speciﬁc costs. In

these studies, the authors suggest that players play a

modiﬁed game with cost functions potentially differ-

ent than the actual game, without considering either

misinformation or coordination mechanisms.

3 SINGLE COMMODITY

NON-ATOMIC CONGESTION

GAMES

We deﬁne single-commodity non-atomic congestion

games using an approach similar to that of (Meir and

Parkes, 2015b):

Deﬁnition 1. A single-commodity non-atomic con-

gestion game is a tuple Γ = hG,l, s,t,ri, where:

• G = (V,E) is a directed graph,

• l is the set of non-decreasing, continuous, and

non-negative latency cost functions with l

(x) :

≥0

→ R

≥0

(one for each edge e ∈ E), where x

is the ﬂow,

• s ∈ V is the source,

• t ∈ V is the destination,

• r is the total mass of ﬂow.

We consider that any player controls an inﬁnitesi-

mal amount of ﬂow r.

Let P be the set of total paths from s to t, then

we deﬁne as f (r) ∈ [0,r]

|P|×1

a feasible ﬂow of the

players routing r units of ﬂow on the paths. We deﬁne

as f

(r) the ﬂow of players that follows the path p ∈

P. The ﬂow of the players on the edge e ∈ E is f

(r).

The cost of following a path p is C

( f

(r)) =

∑

e∈p

( f

(r)) and the total Social Cost of the ﬂow

f (r) is SC( f (r)) =

∑

e∈E

(r)l

( f

(r)). The so-

cially optimal ﬂow is the feasible ﬂow f (r) such that

SC( f (r)) is minimum. We deﬁne a pure Nash equi-

librium (also known as Wardrop equilibrium) in non-

atomic congestion games as follows:

Deﬁnition 2 (Pure Nash equilibrium (Nash,

1951)/Wardrop equilibrium (Wardrop, 1952)).

A pure Nash equilibrium is a feasible ﬂow f

∗

(r) such

that for any p, ˆp ∈ P,

( f

∗

(r)) ≤ C

ˆp

( f

∗

ˆp

(r)),

in other words, a ﬂow is an equilibrium if no player

has any incentive to deviate from her path.

Any non-atomic game has at least one equilibrium

(Schmeidler, 1973; Rosenthal, 1973).

Next, we present the speciﬁc class of single-

commodity non-atomic congestion games with n par-

allel links from s to t with r = 1. We consider that

the cost function of an edge/link k ∈ {1,2,...,n} is

an afﬁne function, that is, l

) = a

+ b

, with

> 0 , and x

∈ [0,1] is the ﬂow of link k, with

∑

k=1

= 1. Without loss of generality, we assume

that the links are sorted in an increasing order with

respect to b

, or in other words that if k ≤ p, then

≤ b

. Let x = (x

,...,x

) be an allocation of the

ﬂow on the links. Here the cost on an edge k is

), thus the total Social Cost of this allocation is

∑

k=1

). We denote this game as Γ, see Figure 1.

n−1

Figure 1: Single-commodity non-atomic congestion game

with n parallel links.

In the case of non-atomic congestion games with

n parallel links, Deﬁnition 2 provides that an allo-

cation x

∗

= (x

∗

,...,x

∗

) of the ﬂow on the links is a

pure Nash equilibrium if and only if, ∀ i ∈ {1,...,n},

and ∀ j ∈ {1,...,n}, l

∗

) ≤ l

∗

). Consequently,

∗

= (x

∗

,...,x

∗

) is a pure Nash equilibrium if and

only if, l

∗

) = l

∗

) for all i, j ∈ {1,...,n} with

∗

> 0. Moreover, all Nash equilibria result in the

same Social Cost (Chapter 18 of (Nisan et al., 2007)).

Coordination Mechanisms with Misinformation

239

4 MISINFORMATION

NON-ATOMIC CONGESTION

GAMES

Misinformation games were deﬁned in (Varsos et al.,

2021) for the general class of normal-form games.

Under the deﬁnition of (Varsos et al., 2021), a misin-

formation game of N players consists of N + 1 game

speciﬁcations (mG = hG

,...,G

N+1

i). The ﬁrst

) is the so-called actual game, which corresponds

to the game that the players actually play, whereas

each one of games G

corresponds to the subjective

view of the game from the perspective of player i, that

is the game that player i believes that is being played.

Note that this deﬁnition allows all types of mis-

information to occur (including, e.g., cases where

the players are misinformed regarding the available

strategies or players). However, in (Varsos et al.,

2021) a special form was considered, called canonical

misinformation games, where subjective views differ

only in the values of the elements of the payoff matri-

ces. It was shown that all non-canonical misinforma-

tion games can be transformed into canonical ones,

therefore we can, without loss of generality, restrict

our attention to canonical games.

In this Section, we use an analogous approach

to deﬁne misinformation games for the case of non-

atomic congestion games, where each player has a

subjective view about the game she plays, which may

be different from the others. Formally:

Deﬁnition 3 (Misinformation game). A misinforma-

tion non-atomic congestion game mΓ with θ splitting

is an (N + 1)-tuple mΓ = hΓ

,Γ

,...,Γ

i, where N is

the number of different views of the game that differ-

ent players may assume, Γ

= hG,l,s,t, ri is the ac-

tual game, Γ

= hG,l

,s,t, r

i are the different sub-

jective game speciﬁcations assumed by the players,

of which each player assumes only one, and θ =

hθ

,...,θ

i, where θ

is the portion of players that

experience view Γ

Here, we assume that the total mass of ﬂow across

all Γ

(for j > 0) is equal to the respective mass in

, r

= r. Further, it must hold that

∑

i∈[N]

= 1.

Thus, the players have the correct view of the graph

and the ﬂow at hand, although they may assume dif-

ferent cost functions. In this case, we call the misin-

formation single commodity non-atomic congestion

games as canonical, similar to canonical misinforma-

tion games, as deﬁned in (Varsos et al., 2021).

Deﬁnition 4 (Misinformed equilibrium strategy). A

misinformed strategy is a ﬂow for portion θ

, that θ

with subjective view Γ

follows in a pure Nash equi-

librium strategy of its game view Γ

Deﬁnition 5 (Natural Misinformed equilibrium). A

natural misinformed equilibrium is a ﬂow f such

that each portion θ

plays a misinformed equilibrium

strategy according to its game-speciﬁc view j.

Since any non-atomic congestion game has at least

one Nash equilibrium, then any misinformation game

of a non-atomic congestion game as deﬁned above has

at least one natural misinformed equilibrium.

Having at hand the formal deﬁnition of the natural

misinformed equilibrium, we measure the deteriora-

tion/leverage in efﬁciency of a non-atomic congestion

game due to misinformation. For that, we adapt the

concept of Price of Misinformation (PoM), that was

deﬁned in (Varsos et al., 2021), in the case of non-

atomic congestion games. Formally:

Deﬁnition 6 (Price of Misinformation). Given a mis-

information congestion game, the Price of Misinfor-

mation (PoM) is deﬁned as

PoM =

max

NME

∈NME

SC( f

NME

)

SC( f

opt

)

, (1)

where f

opt

is the ﬂow that minimizes the Social Cost

in the actual game Γ

and the nominator is the worst

(maximum) value of the Social Cost of the set NME

as computed with regards to the actual game.

We can show the following:

Proposition 1. For every misinformation non-atomic

congestion game, we have that:

1 ≤ PoM ≤ (r · max

p∈P

(r)) /opt (2)

Proof. In the worst case, all ﬂow is routed through the

most costly routes, which leads to a Social Cost of r ·

max

p∈P

(r). Thus, PoM ≤ (r · max

p∈P

(r)) /opt.

Also, PoM ≥ 1 by deﬁnition.

Note that when Γ

= Γ

∀ j, then PoM coincides

with the Price of Anarchy (PoA). Using the deﬁnition

of PoA and (1), we can link these formulas as follows:

PoM = PoA ·





max

NME

∈NME

SC( f

NME

)

max

∈NE

SC( f

)





, (3)

where NE is the set of all Nash equilibria of Γ

Interesting results can be derived by comparing

the worst Nash equilibrium of Γ

(or PoA of Γ

) with

the worst natural misinformed equilibrium of mΓ (or

PoM of mΓ). If PoM < PoA, then misinformation has

a beneﬁcial effect on the social outcome, as players

are inclined (due to their misinformation) to choose

socially better strategies. Otherwise (PoM > PoA),

misinformation leads to a worse outcome from the

perspective of social welfare. Next, we provide an

illustrative example of the above concepts.

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

240

Example 1. We consider the non-atomic congestion

game as depicted in Figure 2a (known as Pigou net-

work (Pigou, 1933)), with latency functions l

(x) =

εx + 1, l

(x) = x + ε, r = 1 and x ∈ [0,1]. It is clear

that selﬁsh players in a pure Nash equilibrium will all

choose route r

, resulting to a Social Cost equal to

1 + ε ≈ 1, for arbitrarily small ε > 0. On the other

hand, the social optimum is achieved by allocating

the ﬂow as follows: ≈

/2 through route r

and ≈

through route r

; so, SC( f

opt

) ≈

/4, and PoA ≈

/3.

Now, consider the actual game Γ

as depicted

in Figure 2a and the game Γ

as depicted in Fig-

ure 2b. Also, assume the misinformation game mΓ =

hΓ

,Γ

i with θ splitting, where Γ

= hG,l,s,t,ri,

= hG,l

,s,t, r

i, Γ

= hG,l

,s,t, r

i, and θ =

hθ

,θ

i. Further, θ

/3 of the players have the

view Γ

, and the rest the view Γ

. In this example,

= l

= l, l

(x) = εx + 1, and l

(x) = x + 1. The

equilibrium for Γ

, Γ

is to choose the route r

, r

re-

spectively, leading to SC

NME

( f

NME

) ≈

/9 in the mΓ;

so PoM ≈

/27 < PoA. Thus, the social outcome is

improved, despite selﬁshness and misinformation.

A metric similar to PoM was introduced in (Meir

and Parkes, 2015b), called the Biased Price of Anar-

chy, which measures the ratio of the equilibrium un-

der biases in knowledge compared to the optimal out-

come. In this concept, all players play a game with

modiﬁed costs and, thus, (possibly) different than the

actual costs. In our concept, all players play a game

according to the misinformation that they assume that

is the same for anyone resulting to different outcomes,

so in general the two concepts PoM and Biased Price

of Anarchy are not equal. The following example clar-

iﬁes the differences between the two studies:

Example 2. Consider now the game as provided in

Figure 2a, where

/2 of the players knows the latency

functions: l

(x) = x and l

(x) =

/4, whereas the rest

knows l

(x) = x+1 and l

(x) = 1, for routes r

and r

respectively. The equilibrium point provided by (Meir

and Parkes, 2015a) is that the ﬂow is splitted into two

equal halfs, each routed through different edge.

On the other hand, the misinformation non-

atomic game that is produced takes the form mΓ =

hΓ

,Γ

i with θ splitting, θ = h

/2,

/2i. The ﬂow

in Γ

has Nash equilibrium such that

/4 is routed

through r

, while

/4 is routed through r

. Similarly,

in Γ

, the Nash equilibrium is such that all the ﬂow

is routed through r

. Therefore, the resulting natural

misinformed equilibrium is

/8 routed through r

, and

/8 routed through r

5 THE WATERFILLING

ALGORITHM

Here, we provide an algorithm that computes a pure

Nash equilibrium in a single-commodity non-atomic

congestion game with n parallel links, and afﬁne la-

tency functions, inspired by the waterﬁlling theorem

of Information theory (Cover and Thomas, 2006; Fa-

soulakis et al., 2019). To our best knowledge, there is

not a similar approach in the literature.

One of the fundamental problems in wireless com-

munications is how to allocate a budget of power

in a constant number of different quality (different

noise levels) and independent wireless communica-

tion channels in order to maximize the sum of the

transmission rate. The optimal solution of this prob-

lem is given by the waterﬁlling theorem. Namely,

the algorithm ﬁlls with water (power) the channels

in a way that minimizes the maximum level of wa-

ter, where the level of water is the maximum value of

power plus noise in the channels that are used. By the

end of the algorithm, the noise plus the water in the

channels that are used is the same (see Figure 3)

Interestingly, this idea can be used to ﬁnd a Nash

equilibrium ﬂow allocation in a single-commodity

non-atomic congestion game with n parallel links and

afﬁne latency functions.

Algorithm 1: Waterﬁlling approach algorithm for comput-

ing a pure Nash equilibrium in single-commodity non-

atomic congestion games with n parallel links and afﬁne

latency functions.

Input: n parallel links with afﬁne latency

functions a

+ b

, with a

> 0,

∀ k ∈ {1,2,...,n}.

Output: A pure Nash equilibrium allocation.

Sort links in an increasing order based on b

for 1 ≤ i ≤ n do

Solve the Linear program (Algorithm 2), for

t = b

i+1

and j = i. If it returns a feasible

solution x

∗

, then STOP and return x

∗

By the deﬁnition of Nash equilibrium, we know

that there is at least one Nash equilibrium in which all

links that are used have the same latency, v = a

and any link k that is not used has a latency no less

than v, or in other words a

· 0 + b

≥ v. However,

we do not know a priori the value of v, but we do

know that the possible values are between the inter-

vals of [b

],(b

],...,(b

n−1

],(b

,+∞), since

s are sorted in an increasing order. Note that if

the optimal threshold is in interval [b

k−1

], then in

the Nash equilibrium allocation we will have exactly

k −1 links. We exhaustively search the optimal values

Coordination Mechanisms with Misinformation

241

(x)

(a) Players without misinformation.

(x)

(b) Misinformed players.

Figure 2: Non-atomic congestion game with 2 parallel links.

in an increasing order for any possible interval, see

Algorithm 1. For any interval we solve a linear pro-

gram to check if there is an allocation with the Nash

equilibrium properties, if there is such an allocation

we return it. Since, we know the existence of such

an equilibrium, our algorithm always returns a feasi-

ble solution at the end. It is easy to see that the total

computational time of the algorithm is polynomial.

Algorithm 2: Linear program.

Input: A positive threshold t and an index j.

Output: x.

minimize v

s.t. x

v−b

, for i ≤ j.

∑

i∈[n]

= r.

≥ 0,v ≥ 0 and v ≤ t.

6 COORDINATION

MECHANISMS WITH

MISINFORMATION

Now we focus on single-commodity misinformation

non-atomic congestion games, where the actual game

has n parallel links and afﬁne latency functions. We

restate the main question of the paper, that is how we

can use misinformation to improve the performance

of single-commodity non-atomic congestion games

with n parallel links in terms of Social Cost.

Speciﬁcally, note that if we properly change the

coefﬁcients of the latency functions of the misin-

formed games then the ﬂow according to the worst

natural misinformed equilibrium will change. To that

direction we choose to increase the coefﬁcients of the

latency function from a

to ˆa

(one for each dif-

ferent subjective view Γ

respectively). Further, we

assume that the designer has the constraint that he can

provide a limited number of misinformed views.

We will show that it is always possible to ﬁnd

a unique natural misinformed equilibrium that co-

incides with the optimal allocation, in terms of so-

cial welfare, by appropriately changing the coefﬁ-

cients. The constructed misinformation game mΓ =

hΓ

,Γ

,...,Γ

i, with θ splitting, described in the next

Subsection, has the following properties:

i) Γ

= Γ (the case of n parallel links),

ii) Γ

= hG, l,s,t,1i, Γ

= hG, l

,s,t, 1i, where

) = a

+ b

, l

) = ˆa

iii) θ = hθ

,...,θ

Next, we provide a methodology to construct a

misinformation game so that the unique natural mis-

informed equilibrium in the misinformation game is

an optimal allocation of the actual game. Forthwith,

we give an algorithm that takes as input a single-

commodity non-atomic congestion game Γ

, the op-

timal allocation in Γ

, and an arbitrary break down

(partition) of the links that are used in the optimal

ﬂow. Let x

∗

= (x

∗

,...,x

∗

) be the optimal solution,

which we can easily ﬁnd in polynomial time as a min-

imization of a convex function, and an abstract parti-

tion (k

,...,k

) of the allocation x

∗

over the parallel

links that are used; m is the number of misinformed

views that the designer provides to the players. E.g.,

if n = 3 with x

∗

> 0 ∀ i, then k

= {1,2}, k

= {3} is

a possible partition. With Algorithm 3 we construct a

misinformation game mΓ, where players perform op-

timally in terms of Social Cost.

To produce mΓ in Algorithm 3, we call Algo-

rithm 4 to compute the coefﬁcients for the latency

functions for each Γ

separately. Afterwards, mΓ is

entailed easily.

At the beginning of Algorithm 4, we initialize v

by setting it equal to the maximum of the costs of the

latency functions over the links that are used in the

allocation y

∗

. Then, we increase the b

for the unused

links in y

∗

in order to make them no less than the cost

v. For any link i that is used in the allocation y

∗

can increase the b

in such a way that the cost of this

link with the allocation y

∗

is equal to the cost v. This

procedure can be done in polynomial-time by solving

a system of linear inequalities. For any y

∗

it is easy to

see that Algorithm 1 gives a unique pure Nash equi-

librium in the modiﬁed game for the players that have

this view. Taking the natural misinformed equilibrium

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

242

Transmitter

Noise

Figure 3: Two possible cases of waterﬁlling with different budget of power.

Algorithm 3: Coordination mechanism algorithm for

an abstract partition (k

,... ,k

) of n.

Input : An actual game Γ

= hG,l

,s,t, 1i

An optimal allocation x

∗

= (x

∗

,...,x

∗

)

A partition over the links, (k

,...,k

Output: A misinformation game mΓ.

A splitting θ.

while 1 ≤ i ≤ m do

New allocation y

∗

(

∗

∑

t∈{k

}

∗

, j ∈ k

0, elsewhere

Apply Algorithm 4 for y

∗

construct the latency functions of Γ

∑

j∈{k

}

∗

mΓ ← hΓ

,Γ

,...,Γ

θ ← hθ

,...,θ

Algorithm 4: Coordination mechanism algorithm.

Input : Coefﬁcients a

of latency functions

for any link i. The allocation y

∗

Output: New coefﬁcients of latency

functions.

Put v = max

i:y

∗

+ b

Find

, ∀ i,

s.t.

≥ v, ∀ i such that y

∗

= 0,

∗

= v −

, ∀ i such that y

∗

> 0,

≥ b

, ∀ i.

return

we construct the allocation x

∗

, which is the optimal

allocation of the actual game, hence PoM = 1.

A similar mechanism can be used when the de-

signer can inﬂuence only some of the players, i.e.,

when she can construct a mechanism where the θ

portion gets misinformed, whereas the rest use the ac-

tual game. In this scenario, the resulting misinforma-

tion game would be of the form mΓ = hΓ

,Γ

where Γ

= Γ

, for the splitting θ = hθ

,θ

To do so, we reconsider the optimal allocation for

the θ

fragment of the ﬂow, taking into account the

fact that there is a ﬁxed part of the players θ

who

will route according to Γ

, which has the same la-

tency functions as Γ

. We reconsider the coefﬁcients

of the latency functions as they experience the addi-

tional cost of the Nash equilibrium ﬂow of the unmis-

informed fragment θ

. Afterwards, we implement our

mechanism and get the desired mΓ. Note that, as we

can affect only some of the players, the rest would

route (maybe sub-optimally) according to the actual

speciﬁcations, so it may happen that PoM > 1.

7 CONCLUSIONS

In this paper, we explored the use of misinforma-

tion as a novel method for coordination mechanisms.

We applied this idea in single-commodity non-atomic

congestion games with parallel links and afﬁne la-

tency functions. Our goal was to steer players’ be-

haviour towards the socially optimum allocation, by

misinforming them with regards to the latency func-

tions of the network. Towards this, we provided two

polynomial-time algorithms. The ﬁrst ﬁnds a Nash

equilibrium ﬂow allocation in a single-commodity

non-atomic congestion game with n parallel links and

afﬁne cost functions. The second takes as input an ab-

stract partition over the links that are used in the op-

timal allocation, and creates a misinformation game

whose subjective games follow the required speciﬁ-

cation, and, thus, its natural misinformed equilibrium

is the optimum allocation in the actual game. Further,

we redeﬁne misinformation games (originally pro-

posed for normal-form games (Varsos et al., 2021))

for non-atomic congestion games and recast all rele-

vant game-theoretic concepts for the new setting.

A future step is to design a mechanism for serial-

parallel networks and general latency functions.

ACKNOWLEDGEMENTS

The second author was supported by the Stavros

Niarchos-FORTH postdoc fellowship for the project

ARCHERS.

Coordination Mechanisms with Misinformation

243

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