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Journal of Quantitative Economics
Published in: Journal of Quantitative Economics 4/2019

16-04-2019 | Original Article

Unilateral Support Equilibrium, Berge Equilibrium, and Team Problems Solutions

Author: Bertrand Crettez

Published in: Journal of Quantitative Economics | Issue 4/2019

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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 agent non-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.
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Footnotes
1
In French, “point d’équilibre pour P relativement à un ensemble K de joueurs.”
 
2
Courtois et al. (2018) call this equilibrium a coalitional Berge equilibrium.
 
3
A presentation and a history of Berge equilibrium can be found in Collman et al. (2011) as well as in Courtois et al. (2015). A recent survey of this notion is proposed by Larmani and Zhukovskii (2017).
 
4
We shall always assume that the maximum and the minimum are realized.
 
5
Studies of Berge–Vaisman equilibrium can be found in Collman et al. (2011), Larbani and Nessah (2008), Musy et al. (2012), Crettez (2017a) and Courtois et al. (2018).
 
6
Existence results for the notions of Berge equilibrium, Berge–Vaisman equilibrium, Berge–Nash equilibrium and coalitional Berge equilibrium can be found in Courtois et al. (2018).
 
7
Of course, the donnees usually do not give to the donators. To account for this fact we may assume that the payoff functions of the donators do not depend on the decisions of the donnees.
 
8
Saflaty and Abdou (2018) use tensors to study the relationship between unilateral support equilibria, Nash and Berge equilibria.
 
9
In this paper we disregard the possibility that agents base their actions on different information sets. Heterogeneity in information is a key assumption in Team theory (see Radner 1991, for a quick presentation of Team theory).
 
10
See also Başar (2017).
 
11
A function \(f : D \subseteq \mathbb {R}^n \rightarrow \mathbb {R}\) is pseudoconcave if for all \(\mathbf {x}_{0}\) and \(\mathbf {x}\) in D, \(\nabla f(\mathbf {x}_{0})(\mathbf {x}-\mathbf {x}_{0}) \le 0 \Rightarrow f(\mathbf {x}) \le f(\mathbf {x}_{0})\).
 
12
The parameter \(\rho \) represents a tipping point of the game.
 
13
In this inequation \(U_k(\mathbf {s}^*_{G}, \mathbf {s}_{N \setminus G})\) is the utility of player k when all the group members follow the practice while the non-members play the strategy profile \(\mathbf {s}_{N \setminus G}\).
 
14
The notion of strict unilateral support equilibrium parallels that of a strict Nash equilibrium (see, e.g., Peters 2008, p. 118).
 
15
We indeed have for all \(G \not = N\), for all i in G, and for all \(j \not \in G\), \( \min _{ \mathbf {s}_{N \setminus G} \in \Pi _{j \in N \setminus G} S_j, } U_i(\mathbf {s}^u_{G}, \mathbf {s}_{N \setminus G}) \le U_{i}(s^u_{i}, s_{j},\mathbf {s^u}_{- \lbrace i, j \rbrace }) < U_{i}(\mathbf {s}^u)\) for \(s_j \not = s_j^u\).
 
16
In what follows we adapt to unilateral support equilibrium what was done for Berge equilibrium in Crettez (2017b).
 
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Metadata
Title
Unilateral Support Equilibrium, Berge Equilibrium, and Team Problems Solutions
Author
Bertrand Crettez
Publication date
16-04-2019
Publisher
Springer India
Published in
Journal of Quantitative Economics / Issue 4/2019
Print ISSN: 0971-1554
Electronic ISSN: 2364-1045
DOI
https://doi.org/10.1007/s40953-019-00168-w

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