Unilateral Support Equilibrium, Berge Equilibrium,
and Team Problems Solutions
Bertrand Crettez∗
December 21, 2018
Abstract
We compare two notions of equilibrium for other-regarding agents, namely Berge and unilateral
support equilibria. A Berge equilibrium is a strategy profile such that the teammates of each
agent choose their strategies in order to maximize his utility. A unilateral support equilibrium is a
strategy profile such that the teammates of each agentnon-cooperatively choose their strategies to
maximize his utility. By definition the level of cooperation in a unilateral support equilibrium is no
higher than in a Berge equilibrium. Yet, relying on ideas from Team theory, we provide conditions
under which a unilateral support equilibrium is also a Berge equilibrium. We also provide conditions
under which a unilateral support equilibrium is a Berge-Vaisman equilibrium,i.e., a strategy profile
which is a Berge equilibrium and such that the payoff of each player is no lower than his maximin
value.
Key Words : Berge equilibrium, Berge-Vaisman equilibrium, Berge-Nash equilibrium, Unilateral support equilibrium, Team optimal solution, Person-by-person optimal solution, Mutually beneficial
practice.
JEL Classification Numbers : C7, D 74.
∗Universit´e Panth´eon-Assas, Paris II, CRED, EA 7321, 21 Rue Valette - 75005 PARIS,
1
Introduction
Ordinary market transactions do not always rely on self-interest alone, but are often carried out
with goodwill on both sides (Sugden, 2011, 2015, 2018). There is also evidence from experimental
games (as in the Market Game or the Trust Game) that cooperative behavior occurs quite frequently
(Sugden, 2015 ; see also Colman et al., 2008, Gualaet. al., 2013).
To account for these observations one may assume that people are other-regarding. The notion
of Berge equilibrium is a major solution concept for analyzing the interactions of other-regarding
agents (Berge, 1957, pp 88-89 and Zhukovskiy, 1985). A Berge equilibrium is a strategy profile
such that the teammates of each agent choose their strategies in order to maximize his utility. The
purpose of this paper is to further the comparison of Berge equilibrium with unilateral support
equilibrium, an alternative solution concept for other-regarding agents proposed by Schouten et al
(2018). A unilateral support equilibrium is a strategy profile such that the teammates of each agent
non-cooperatively choose their strategies to maximize his utility. This notion seeks to address the
criticism that the level of cooperation in a Berge equilibrium can be unrealistically high.
Our study of the relationships between Berge and unilateral support equilibria notably builds on
Team theory. A team, as Marschak (1955, p. 128) puts it, is a “group of persons each of whom
takes decisions about something different but who receive a common reward as the joint result of
all those decisions.” The optimal solution of the team problem, namely jointly maximizing the
common payoff function, is closely linked to the notion of Berge equilibrium. Similarly, the
person-by-person optimal solution of the team problem, which is nothing but a Nash equilibrium for the
game where the players are the team members, is connected to the notion of unilateral support
equilibrium. We rely on these solution concepts for team problems to analyze the conditions under
which unilateral support and Berge equilibria coincide.
In addition, we provide conditions under which the payoff obtained by each player in a unilateral
support equilibrium is no lower than his maximin value (self-sacrifice is then avoided). Specifically,
we present two sufficient conditions under which a unilateral support equilibrium is a Berge-Vaisman
equilibrium,i.e., a Berge equilibrium such that the payoff of each player is no lower than his maximin
value.
a mutually beneficial practice. A mutually beneficial practice for an n-player game is a strategy
profile which satisfies two conditions. First, each agent gain must be higher than his maximin
payoff. Second, where a group of players follows the practice there must be a group member whose
payoff is higher with the practice than without: cooperation is beneficial for at least one agent. We
already know some conditions under which a Berge equilibrium is a mutually beneficial practice.
We extend these conditions to the case of a unilateral support equilibrium.
The remaining part of this paper unfolds as follows. In the next section we present the setup and
formal definitions. Thereafter, section 3 compares unilateral support equilibrium, Berge equilibrium
and solutions to team problems. In section 4, we focus on the case where a unilateral support
equilibrium is such that the payoff for each agent is no lower than his maximin value. Section 5
considers the relationships between unilateral support equilibrium and mutually beneficial practice.
Section 6 briefly concludes the paper.
2
Model and Definitions
Consider an n-player gameG≡ hN,(Si)i∈N,(Ui)i∈NiwhereN ={1, . . . , n}is the set of players,Si
is the strategy set of playeri, andUi: Πi∈NSi →Ris his utility function. Denote bysa strategy
pro-file (s1, s2, . . . , sn) in Πi∈NSiand bys−ithe incomplete strategy profile (s1, s2, . . . , si−1, si+1, . . . , sn)
in Πj6=iSj. Also denote byUi(si,s−i) the image ofs by Ui,i.e.,Ui(s1, s2, . . . , si−1, si, si+1, . . . , sn).
We next introduce various notions of cooperation for others-regarding agents.
2.1 Berge Equilibrium
Let us first recall the definition of Berge equilibrium.
Definition 1. A strategy profile s is a Berge equilibrium if
Ui(si,s−i)≤Ui(s), ∀s−i ∈Πj6=iSj, ∀i∈N. (1)
In a Berge equilibrium each agent is supported by his teammates in the sense that all the other
agents maximize his utility. This definition calls to mind the behavior rule of Alexandre Dumas’s
Berge equilibrium is sometimes called Berge-Zhukovskiy equilibrium. By using the term Berge
equilibrium we follow Courtois et al. (2015) as well as Larmani and Zhukovskiy (2017). Berge
(1957, p. 88-89) actually refers to P/K-equilibrium.1 A P/K- equilibrium is obtained when the
players belonging to the coalition K maximize the payoffs of the players belonging to the coalition
P.2 The notion of Berge equilibrium proposed by Zhukovskiy (1985) is a special case of P/K
-equilibrium, where P includes one player andK includes all the other ones.3
Two refinements of Berge equilibrium address the issue of individual rationality: Berge-Vaisman
equilibrium and Berge-Nash equilibrium. We consider these refinements in turn.
Berge-Vaisman equilibrium
Let αi be the maximin security level of player i: αi ≡ maxsi∈Simins−i∈S−iUi(si,s−i).
4 This is
the gain that a player can guarantee even if all the other players jointly try to minimize his payoff
function. It is also a benchmark for defining the benefits of cooperation (cooperation is fruitful
only in so far as it brings about a gain higher than the maximin value).
The cooperation achieved in a Berge equilibrium can be jeopardized if the maximin gain of one
player is higher than his equilibrium payoff. In that case, this player would be certain to obtain a
better payoff by deviating unilaterally. This remark led Zhukovskiy and Chikrii (1994) to introduce
the notion of Berge-Vaisman equilibrium. A Berge-Vaisman equilibrium is a Berge equilibrium such
that the payoff of each player is no lower than his maximin gain.5
Definition 2. A strategy profiles∗ is a Berge-Vaisman equilibrium if it is a Berge equilibrium and if it also satisfies the following individual rationality condition:
αi ≤Ui(s∗), ∀i∈N. (2)
Berge-Nash equilibrium
1
In French, “point d’´equilibre pourP relativement `a un ensembleKde joueurs.” 2
Courtoiset al. (2018) call this equilibrium a coalitional Berge equilibrium. 3
A presentation and a history of Berge equilibrium can be found in Collmanet al. (2011) as well as in Courtois
et al. (2015). A recent survey of this notion is proposed by Larmani and Zhukovskiy (2017). 4We shall always assume that the maximum and the minimum are realized.
5Studies of Berge-Vaisman equilibrium can be found in Colemanet al.(2011), Larbani and Nessah (2008), Musy
This notion, proposed by Abalo and Kostreva (2004), merges the definitions of Berge and Nash
equilibria. Of course, the level of individual rationality in a Berge-Nash equilibrium is higher than
in the notion of Berge-Vaisman equilibrium. In addition, observe that a Berge-Nash equilibrium is
a Berge Vaisman equilibrium but the converse is not always true.6
2.2 Unilateral support equilibria
The notion of unilateral support equilibrium, introduced in Schoutenet al. (2018), builds on that
of Berge equilibrium but embodies the idea of individual support as opposed to that of collective
support. It seeks to avoid the criticism that Berge equilibrium may require an unlikely high level
of cooperation.
Definition 3. A unilateral support equilibrium is a strategy profile su such that for each pair of different players iand j in N
Ui(sui, sj,su−{i,j})≤Ui(su),∀sj ∈Sj. (3)
Schoutenet al. (ibid) show that the set of unilateral support equilibria is exactly the intersection of
the sets of unilaterally supportive strategy profiles with respect to all derangement. A derangement
is a bijection defined, and taking values, on the set of players, which never maps a player to himself.
A strategy profile is unilaterally supportive if each player maximizes the payoff of the individual to
whom he is mapped. Actually, a strategy profile is unilaterally supportive if it is a Nash equilibrium
where each player ihas agent’sj objective.7
Schouten et al. also give an example of a three-person matrix game in which there exists a
uni-lateral support equilibrium but no Berge equilibrium. Furthermore, they show that every Berge
equilibrium is a unilateral support equilibrium (Theorem 3.5), and that a unilateral support
equi-librium is always a Berge equiequi-librium when the number of players is equal to 2 (Proposition 4.2). In
addition, they consider what they call coordination games, in which all the players have the same
payoff function. In this setting, they show that the set of unilateral support equilibria coincides
with the set of Nash equilibria (Proposition 4.5). Finally, for general games they show that every
6
Existence results for the notions of Berge equilibrium, Berge-Vaisman equilibrium, Berge-Nash equilibrium and
coalitional Berge equilibrium can be found in Courtoiset al. (2018).
7Saflaty and Abdou (2018) use tensors to study the relationship between unilateral support equilibria, Nash and
strategy profile that is both a unilateral support equilibrium and a Nash equilibrium is also a Nash
equilibrium of a game in which every player faces the payoff function of a single player (Theorem
4.6).
As will be apparent, there are some connections between Berge equilibrium, unilateral support
equilibrium and the solution concepts for team problems.
2.3 Team, team optimal solution, and person-by-person optimal solution
In a team problem, as in a coordination game, all teammates have the same payoff function, i.e.,
Ui =U for all iinN, although they may have different strategy sets.8
Definition 4. A Team optimal solution is obtained when all agents maximize their common payoff function in a cooperative way. Formally, they solve the next team problem:
max
s∈Πi∈NSiU(s). (4)
Yet, we may also assume that the team members try to maximize their common payoff in a
non-cooperative way. In this connection, following Radner (1962)9 we define aperson-by-person optimal
solution for an n-player game as follows.
Definition 5. A person-by-person optimal solution for an n-player game is a strategy profile
(s1, . . . , sn) such that for all i in N we have
U(si,s∗−i)≤U(s∗i,s∗−i), ∀si ∈Si. (5)
It is immediate to see that a person-by-person optimal solution is nothing but a Nash equilibrium
of a game in which all the players have the same payoff function.
The next section compare the different solution concepts presented above.
8
In this paper we disregard the possibility that agents base their actions on different information sets.
Hetero-geneity in information is a key assumption in Team theory (see Radner, 1988, for a quick presentation of Team
theory).
3
Comparisons of Berge Equilibrium, Unilateral Support
Equilib-rium and Team Solutions
We begin by comparing team optimal solutions and Berge equilibria.
Proposition 1. Any team optimal solution is a Berge equilibrium.
Proof. By Definition 5, a strategy profiles∗ is a team optimal solution whenever
U(s)≤U(s∗),∀s∈Πi∈NSi. (6)
It then follows that
U(s∗i,s−i)≤U(s∗),∀s−i ∈Πj∈N
j6=i Sj, (7)
which proves that s∗ is a Berge equilibrium.
As the following example shows, the converse result is false.
Example 1. Assume that the common utility function isU(s) = min{s1, s2, s3}−θmax{s1, s2, s3}, where 0 < θ < 1. Also assume that si ∈ {s,s, β¯ s¯}, i = 1,2,3, where s < s¯and β > 1. We can
check thatsi=s,i= 1,2,3, is a Berge equilibrium. The same can be said for the strategy profiles
such that si = ¯s for all i, and si = βs¯for all i. In this last case, the strategy profile is also the
team optimal solution. But we see that there are two Berge equilibria which differ from the team
optimal solution.
As was mentioned above, a Berge equilibrium is a unilateral support equilibrium. It then follows
from Proposition 1 that a team solution is also a unilateral support equilibrium. We know,
how-ever, that a unilateral support equilibrium is not necessarily a Berge equilibrium, and thus, is not
necessarily a team optimal solution. To further the comparison of Berge and unilateral support
equilibria we now rely on team solutions for team problems.
Observe, first, that any Berge equilibrium embodiesnteam optimal solutions tonteam problems.
That is because, for each agentithen−1 other players maximize thesame function, namely agent i’s payoff functionUi(sbi,s−i) in the set Πj6=iSj. Also observe, second, that any unilateral support
n−1 players maximize his payoff function, albeit in a non-cooperative way. The next result follows directly from these observations.
Proposition 2. A unilateral support equilibrium is a Berge equilibrium if, and only, if, for each player i in N the associated team-optimal and person-by-person optimal solutions are identical.
Therefore, comparing Berge and unilateral support equilibria boils down to comparing n team
optimal solutions with n corresponding person-by-person optimal solutions. Yet, we know from
Team theory that there are instances where team optimal solutions and person-by-person optimal
solutions coincide. In particular, Radner (1962) showed that if the utility function U(.) of a team
problem is concave and differentiable any (interior) person-by-person optimal solutionis a solution
to the team problem. We can rely on this fact to prove the next result.
Proposition 3. Suppose that for all agent i in N the set Πj6=iSj is convex and that the payoff
functionUi(.) is concave and differentiable. Let a unilateral support equilibrium su be given. Then,
if su−i is in the interior of Πj6=iSj for alli in N, su is a Berge equilibrium.
Proof. Consider a unilateral support equilibrium su such that the assumption of the Proposition hold. Since for alliinN and for allj 6=i,Ui(sui, sj,su−{i,j})≤Ui(su) andsu−i is an interior solution,
we must have
∂Ui
∂sj
(su) = 0. (8)
Moreover, since Ui is concave and differentiable we get (see, e.g. Ok, 2007, Proposition 6, p. 706)
Ui(sui,s−i)−Ui(sui,su−i)≤ ∇−iUi(su). (sui,s−i)−su
= 0,∀s−i∈Πj6=iSj, (9)
where∇−iUi(su) is the vector including all the partial derivatives ofUi, except the one with respect
tosi. Since the conclusion is true for alli, the result follows.
Recall that Schootenet al. (2018, proposition 4.2) establish that a unilateral support equilibrium
is a Berge equilibrium when the number of players is equal to 2. The above Proposition shows that
this condition is not necessary.
As the next Proposition illustrates, there are other sufficient conditions under which a unilateral
Proposition 4. Assume that Ui(.) is differentiable on Πj=6 iSj for all i in N. Let an interior
unilateral support equilibrium su be given.
a) If for all i in N, the team problem maxs−iUi(s u
i,s−i) has a unique optimal solution which is
interior inΠj6=iSj, and the system of equations ∂Ui∂sj(sui,s−i) = 0 has a unique solution given bysu−i,
thensu is a Berge equilibrium.
b) If the function Ui(sui, .) is pseudoconcave on Πj=6 iSj for all i in N, then su is a Berge
equilib-rium.10
Proof. As for assertion a), by definition of a unilateral support equilibrium we have ∂Ui∂sj(su) = 0
for all j 6= i. But these conditions must also be satisfied by a team optimal solution. Since the first-order conditions have a unique solution, then su−i is the optimal solution for the associated team problem. As this is true for alli, it follows from Proposition1thatsu is a Berge equilibrium. Let us turn to assertionb). By assumption su−i is an interior unilateral support equilibrium. Nec-essarily ∂Ui∂sj(su) = 0 for alliandj,j6=i. SinceUi(sui, .) is pseudoconcave,su−i is the team optimal
solution. But since this is true for all iwe conclude again from Proposition 1 that su is a Berge equilibrium.
The following corollary directly results from the above results.
Corollary 1. Consider a coordination game. Let su be an interior unilateral support equilibrium inΠi∈NSi and suppose that the common payoff function U is differentiable onΠi∈NSi and satisfies
one of the following assumptions.
1. U is concave on Πi∈NSi and Πi∈NSi is convex.
2. The problem maxs∈Πi∈NSiU(s) has a unique interior solution and for all i in N the system
of equations ∂sj∂U(su
i,s−i) = 0 (j6=i) has a unique solution given by su−i.
3. The function U(.) is pseudoconcave onΠi∈NSi.
Then su is both a Berge-Nash equilibrium and a person-by-person optimal solution. 10A functionf:D⊆
Proof. Under each of the above assumptions, Propositions3and4ensure that the unilateral support
equilibrium su is a Berge equilibrium. Moreover, by Proposition 4.5 of Schouten et al. (2018) this Berge equilibrium is also a Nash equilibrium and then a person-by-person optimal solution.
4
Unilateral Support Equilibrium and Maximin Payoffs
Recall that a Berge-Vaisman equilibrium is a Berge equilibrium where for each agent the equilibrium
individual payoff is no lower than the maximin gain. This property ensures that the cooperation
obtained in a Berge equilibrium is not jeopardized by a player who is certain to obtain a better
payoff by deviating unilaterally. In the same spirit, we shall call a Vaisman unilateral support
equilibrium a unilateral support equilibrium such that for each player the payoff is no lower than
the maximin gain. Alternatively, we shall say that this equilibrium satisfies the Vaisman property.
There are at least two conditions ensuring that a Berge equilibrium is a Berge-Vaisman equilibrium.
The first condition states that the maximin gain of each player is no higher than the maximum
gain accruing to him if he behaves in a self-destroying way, while all the other players try to
support him (Colman et al., 2011). The second condition states that no equilibrium individual
strategy is strongly dominated (Crettez, 2017, a). When a unilateral support equilibrium is a
Berge equilibrium one can use the two above sufficient conditions to check that the equilibrium
payoffs are no lower than the maximin gains. Yet, as illustrated in the next example, there are
unilateral support equilibria which are not Berge equilibria, but which satisfy the Vaisman property.
Example 2. Consider a three-person game where the common payoff function is
U(s1, s2, s3) =
s1+s2+s3−max{s1, s2, s3},if ρ < s1+s2+s3,
−max{s1, s2, s3},otherwise.
(10)
and si ∈ {s,¯s} ⊂ R+, with s <s.¯11 Assume, furthermore, that 2s+ ¯s < ρ < s+ 2¯s. Then, there is a unilateral support equilibrium where si =s, for all i. We can check that this strategy profile
is not a Berge equilibrium. Also observe that αi =−s¯for alli, and that αi is lower than −s, the
payoff obtained in the unilateral support equilibrium.
This example shows that it would be interesting to establish sufficient conditions directly ensuring
that a unilateral support equilibrium satisfies the Vaisman property. We present two such conditions
next. Taking our inspiration from Courtois et al. (2011), our first condition is given in the next
Proposition.
Proposition 5. Let a unilateral support equilibrium su be given and assume that
αi ≤min
j maxsj minsi Ui(si, sj,s u
−{i,j}),∀i∈N. (11)
Then su is a Vaisman unilateral support equilibrium.
Proof. Suppose that su is not a Vaisman unilateral support equilibrium. Then there is an agenti
for whomUi(su)< αi. But this implies that
Ui(su)< αi≤βi= min
j6=i maxsj minsi Ui(si, sj,s u
−{i,j}). (12)
Therefore there is an agentj different from iand a strategy sj inSj such that
Ui(su)<max
sj minsi Ui(si, sj,s u
−{i,j}) (13)
≤max
sj Ui(s u
i, sj,su−{i,j}) =Ui(su), (14)
which is a contradiction.
Adapting the approach followed in Crettez (2017, a), one can obtain yet another condition under
which a unilateral support equilibrium satisfies the Vaisman property. To do this, we need the next
definition.
Definition 6. Let a strategy profile s be given. Then agent i’s decision si in Si is j-dominated if
there is another decision s0i in Si such that
Ui(si,s˜j, s−{i,j})< Ui(s0i,s˜j, s−{i,j}),∀s˜j ∈Sj. (15)
In plain English this means that agent i’s decision is j-dominated if he is better-off choosing s0i
whatever agent j decides, assuming that the other agents stick to their decisions s−{i,j}. This
definition is close to the definition of a strictly dominated strategy. In that latter case, however,s0i
would be a better decision against any decision s−i. The next condition is in spirit close to the
Proposition 6. Let a unilateral support equilibriumsu be given. Assume that for alliinN there is an agentj 6=isuch thatsui is notj-dominated.Thensu is a Vaisman unilateral support equilibrium. Proof. Suppose that su is not a Vaisman unilateral support equilibrium. Then for some agent iit holds that
Ui(su)< αi = max si mins−i
Ui(si,s−i) (16)
≤Ui(s0i,s−i),∀s−i∈Πj6=i, (17)
for somes0i inSi. Letj be the integer such thatsui is notj-dominated. From the above inequation
we have
Ui(su)< Ui(s0i, sj,s−{i,j}),∀sj ∈Sj,∀s−{i,j} ∈Πk6=i,jSk. (18)
But by definition of a unilateral support equilibrium, we would have
Ui(sui, sj,su−{i,j})≤Ui(sui, suj,s−{u i,j})< Ui(s0i, sj,su−{i,j}),∀sj ∈ΠSj. (19)
It would then follow thatsui isj-dominated, which is a contradiction.
Building on the two previous Propositions and the results obtained in the section 3 we can also state
a set of sufficient conditions for a unilateral support equilibrium to be a Berge-Vaisman equilibrium:
Corollary 2. Assume that either the assumptions of Propositions 3 or 4 are met. Assume further that either the assumptions of Propositions 5 or 6 hold. Then, a unilateral support equilibrium is
also a Berge-Vaisman equilibrium.
5
Unilateral Support Equilibrium and Mutually Beneficial
Prac-tice
The notion of mutually beneficial practice introduced in Sugden (2015) is another attempt at
capturing the idea that agents often tend to cooperate and gain from cooperating. More precisely,
where agents follow the practice, any agent should expect a decrease in his payoff if a group of
players cease to follow the practice and take an aggressive stance towards him.
For ann-player game, Sugden (2015) proposes the following formal definition of mutually beneficial
Definition 7. In any n-player game, a strategy profile s∗ = (s∗1, . . . , s∗n) is a mutually beneficial practice if
1. αi < Ui(s∗), for all playersi in N,
2. and for all nonempty groups of players G (G 6= N), there is a player k in G such that:
minsN\G∈Πj∈N\GSj,Uk(s
∗
G,sN\G)< Uk(s∗).12
As for the first condition of Definition7Sugden (2015) recalls that the maximin payoff is a
bench-mark for defining the benefits of cooperation.
Condition 2 specifically requires that for any arbitrary group of agents there must exist one member
of this group for whom the gain with the practice isstrictly higher than his (group-based) minimum
payoff. The following definition will be useful.
Definition 8. We say that a unilateral support equilibrium is strict if all the inequations in (3) are strictly satisfied when sj 6=suj.13
Observe that a strict unilateral support equilibrium satisfies Condition 2 of the definition of a
mutually beneficial practice.14 Therefore, for a strict unilateral support equilibrium to be a mutually
beneficial practice, we must make sure that Condition 1 is satisfied. To do this, we can rely on
what we found in the preceding section for a strategy profile to be a Vaisman unilateral support
equilibrium.15
We have:
Proposition 7. Let a strict unilateral support equilibrium su be given and assume that
αi <min
j6=i maxsj minsi Ui(si, sj,s u
−{i,j}),∀i∈N. (20)
Then su is a mutually beneficial practice. 12In this inequationU
k(s∗G,sN\G) is the utility of playerk when all the group members follow the practice while
the non-members play the strategy profilesN\G.
13
The notion of strict unilateral support equilibrium parallels that of a strict Nash equilibrium (see, e.g., Peters
2008, p. 118). 14
We indeed have for all G 6= N, for all i in G, and for all j 6∈ G, minsN\G∈Πj∈N\GSj,Ui(s u
G,sN\G) ≤
Ui(sui, sj,su−{i,j})< Ui(su) forsj6=suj.
15In what follows we adapt to unilateral support equilibrium what was done for Berge equilibrium in Crettez (2017,
Proof. It is similar to that of Proposition5.
Further, to make use of Proposition 6 we need the next definition, which is a slight variation of
Definition6:
Definition 9. Let a strategy profile s be given. Agenti’s decisionsi in Si is weaklyj-dominated if
there is another decision s0i in Si such that
Ui(si,s˜j, s−{i,j})≤Ui(si0,s˜j, s−{i,j}),∀s˜j ∈Sj. (21)
We have:
Proposition 8. Let a strict unilateral support equilibriumsu be given. Assume that for alli, there is an agent j for whom sui is not weakly j-dominated. Then su is a mutually beneficial practice. Proof. It is similar to the proof of Proposition 6.
6
Conclusion
In this paper we have furthered the comparison of Berge and unilateral support equilibria. We have
shown that the link between these two equilibrium notions can be understood by comparingnteam
optimal solutions with n corresponding person-by-person optimal solutions. To wit, a unilateral
support equilibrium is a Berge equilibrium if, and only if, each team optimal solution coincides
with the corresponding person-by-person optimal solution. We have provided sufficient conditions
for this necessary and sufficient condition to be satisfied. Moreover, we have proposed a set of
sufficient conditions under which the payoff of each agent in a unilateral support equilibrium is no
lower than his maximin value. Finally, we have also shown that a unilateral support equilibrium
can be a Sugden’s mutually beneficial practice.
This research can first be extended by studying the conditions under which unilateral support
equi-libria exist. In addition, it would be interesting to use the notion of unilateral support equilibrium
in the areas where Berge equilibrium has already been used (see, e.g., Brooks, 2018).
Abalo, Kokou Y. and Michael M. Kostreva (2004), Some existence theorems of Nash and Berge
equilibria, Applied Mathematics Letters, 17, 569-573.
Ba¸sar, Tamer (2017), Introduction to the theory of games. In T. Ba¸sar and G. Zaccour (Eds.),
Handbook of Dynamic Game Theory, Springer International Publishing, 2017.
Berge, Claude (1957),Th´eorie g´en´erale des jeux `a n personnes[General theory of n-person games],
Paris, Gauthier-Villars. This document is available at Th´eorie des jeux `an personnes.
Brooks, Richard, 2018, Loyalty and Agency in Economic Theory, Working paper.
Colman, Andrew M., Pulford Briony D., and Jo Rose (2008), Collective rationality in interactive
decisions: evidence for team reasoning,Acta Psychologica, 128, 387-397.
Collman, Andrew M., Koerner Tom W., Musy Olivier and Tarik Tazdait (2011), Mutual Support
in Games: Some Properties of Berge Equilibria, Journal of Mathematical Psychology, 55(2), p.
166-175.
Courtois, Pierre, Nessah Rabia and Tarik Tazdait (2015), How to Play Games? Nash versus Berge
Behavior Rules, 2015,Economics and Philosophy, March, 31, 1.
Courtois, Pierre, Nessah Rabia and Tarik Tazdait (2018), Existence and Computation of Berge
Equilibrium and of Two Refinements,Journal of Mathematical Economics, forthcoming.
Crettez, Bertrand (2017) (a), A new sufficient condition for a Berge equilibrium to be a
Berge-Vaisman equilibrium, Journal of Quantitative Economics, 15, 3, 451-459.
Crettez, Bertrand (2017) (b), On Sugden’s “mutually beneficial practice” and Berge equilibrium,
International Review of Economics, 64, 4, 357-366.
Guala, Francesco, Mittone Luigi, and Matteo Ploner (2013), Group membership, Team Preferences
and Expectations, Journal of Economic Behavior and Organization, 86, 183-190.
Safatly, Elias and Joanna E. Abdou (2018), A New Tensor Approach of Computing Pure and Mixed
Unilateral Support Equilibrium, Working paper.
Larbani, Moussa, and Vladislav Iiosifovich Zhukovskii (2017), Berge equilibrium in normal form
static games: a literature review, Izv. IMI UdGU, Volume 49, 80110.
Larbani, Moussa and Rabia Nessah (2008), A Note on the Existence of Berge and Berge-Nash
Marschak, Jabob (1955), Elements for a Theory of Teams,Management Science, 1, (2).
Marschak, Jacob and Roy Radner (1972),Economic Theory of Teams., Yale University Press, New
Haven.
Musy, Olivier, Pottier Antonin and Tarik Tazdait (2012), A New Theorem to Find Berge Equilibria,
International Game Theory Review, 14(1), 1250005-1 - 1250005-10.
Nessah, Rabia and Moussa Larbani (2014), Berge-Zhukovskii Equilibria: Existence and
Character-ization,International Game Theory Review, 16, 4, 1450012-1-1450012-11.
Ok, Efe A. (2007),Real Analysis with Economic Applications, Princeton University Press,
Prince-ton.
Peters, Hans (2008) Game theory. Springer, Berlin
Saflaty, E. and J.E. Abdou (2018), A New Tensor Approach of Computing Pure and Mixed
Uni-lateral Support Equilibria.
Schouten, Jop, Born, Peter and Ruud Hendrickx (2018), Unilateral Support Equilibria. Working
paper, Tilburg University.
Sugden, Robert (2015), Team reasoning and intentional cooperation for mutual benefit,Journal of
Social Ontology, 1.1, pp. 143166.
Sugden, Robert (2018), The Community of Advantage, Oxford University Press, Oxford.
Radner, Roy (1962), Team Decision Problems,The Annals of Mathematical Statistics, 33, (3).
Radner, Roy (1991), Teams, in The New Palgrave, John Eatwell, Murray Migate and Peter
New-man, eds., Volume 4., 613-616. The Macmillan Press limited. London and Basingstoke.
Zhukovskii, Vladislav I. (1985), Some Problems of Non-Antagonistic Differential Games,
