Equilibria with vector-valued utilities and preference information. The analysis of a mixed duopoly

Mármol, Amparo M.; Caraballo, M. Ángeles; Zapata, A.

doi:10.1007/s11238-017-9595-y

Cited by 17 publications

(14 citation statements)

References 23 publications

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“…In this section we introduce notations and definitions and, to make the article self‐contained, we summarize some results that will be applied in the following sections. The results in Theorems 1 and 2 have been established in Mármol, Monroy, Caraballo, and Zapata () for general games in which the agents have different vector‐valued utilities. The games arising when the agents show ORP are special cases in which all the agents have the same vector‐valued utility.…”

Section: Preliminariesmentioning

confidence: 89%

“…Let

r_{j}^{i}

denote the correspondence of best response of agent i in relation to the j ‐th utility component. Theorem (Mármol et al . ) If for all

i \in N

, A i is a nonempty convex compact subset

A^{i} \subseteq R

, and

u_{j}^{i}

is strictly concave in its own action for each

j \in J^{i}

, then the set of equilibria of the game with vector‐valued utilities

G = {{false(A^{i}, u^{i} false)}_{i \in N}}

E (G) = {(a^{1}, \dots, a^{n}) \in \times_{i \in N} A^{i} : {\underset{r}{true}}^{i} (a^{- i}) \leq a^{i} \leq {\overset{r}{true}}^{i} (a^{- i}), i \in N},

where

{\underset{r}{true}}^{i} (a^{- i}) = m i n_{j \in J^{i}} r_{j}^{i} (a^{- i})

, and

{\overset{r}{true}}^{i} (a^{- i}) = m a x_{j \in J^{i}} r_{j}^{i} (a^{- i})

. A similar result characterizes the set of weak equilibria when the assumption of strict concavity of the compo...…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, in Mármol et al . () it is proven that the equilibria of weighted games with nonnegative weights are weak equilibria of the corresponding game with vector‐valued utilities. When information about the preferences of the agents is available, it can be applied to reduce the set of equilibria.…”

Section: Preliminariesmentioning

confidence: 99%

“…For each

i \in N

, define a function,

v_{Λ}^{i}

, with values in

R^{p_{i}}

, given by

v_{Λ}^{i} = B^{i} \cdot u^{i}

. Theorem (Mármol et al . ) Let

G = {{false(A^{i}, u^{i} false)}_{i \in N}}

be a game with vector‐valued utilities such that each A i is a nonempty convex subset of a finite dimensional space and for each i, u i is concave in a i . Then the set of equilibria of the game with preference information Λ coincides with the set of weak equilibria of the game

{{false(A^{i}, v_{Λ}^{i} false)}_{i \in N}}

.…”

Section: Preliminariesmentioning

confidence: 99%

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Agents with other‐regarding preferences in the commons

2016

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We present a unified approach to study the problem of the commons for agents with other‐regarding preferences. This situation is modeled as a game with vector‐valued utilities. Several types of agents are characterized depending on the importance assigned to the components of their utility functions. We obtain the set of equilibria of the game with two types of agents, pro‐social and pro‐self, and some refinements of this set for conservative agents. The most relevant result is that only a pro‐social agent is required to avoid the tragedy of the commons, regardless of the behavior of the rest of the agents.

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Section: Preliminariesmentioning

confidence: 89%

“…Let

r_{j}^{i}

denote the correspondence of best response of agent i in relation to the j ‐th utility component. Theorem (Mármol et al . ) If for all

i \in N

, A i is a nonempty convex compact subset

A^{i} \subseteq R

, and

u_{j}^{i}

is strictly concave in its own action for each

j \in J^{i}

, then the set of equilibria of the game with vector‐valued utilities

G = {{false(A^{i}, u^{i} false)}_{i \in N}}

E (G) = {(a^{1}, \dots, a^{n}) \in \times_{i \in N} A^{i} : {\underset{r}{true}}^{i} (a^{- i}) \leq a^{i} \leq {\overset{r}{true}}^{i} (a^{- i}), i \in N},

where

{\underset{r}{true}}^{i} (a^{- i}) = m i n_{j \in J^{i}} r_{j}^{i} (a^{- i})

, and

{\overset{r}{true}}^{i} (a^{- i}) = m a x_{j \in J^{i}} r_{j}^{i} (a^{- i})

. A similar result characterizes the set of weak equilibria when the assumption of strict concavity of the compo...…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…For each

i \in N

, define a function,

v_{Λ}^{i}

, with values in

R^{p_{i}}

, given by

v_{Λ}^{i} = B^{i} \cdot u^{i}

. Theorem (Mármol et al . ) Let

G = {{false(A^{i}, u^{i} false)}_{i \in N}}

{{false(A^{i}, v_{Λ}^{i} false)}_{i \in N}}

.…”

Section: Preliminariesmentioning

confidence: 99%

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Agents with other‐regarding preferences in the commons

2016

Self Cite

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“…For example, social responsible firms take into account not only their profits, but also a share of consumer surplus, and this additional goal may heavily influence the equilibrium outcomes. Related work about multicriteria strategic models is Mármol et al (2017) and Monroy et al (2016).…”

Section: Introductionmentioning

confidence: 99%

Duopolistic competition with multiple scenarios and different attitudes toward uncertainty

Caraballo

Mármol

2017

Int Trans Operational Res

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In this paper, we address duopolistic competition when the firms have to assess the results of the interaction at different scenarios. Specifically, we consider the case in which the scenarios are identified with several states of nature and, therefore, the firms face uncertainty on their results. The probability of occurrence of the scenarios is unknown by the firms and they make their output decision before uncertainty is resolved. Within this framework, we analyze competition between firms when these firms exhibit extreme and neutral attitudes toward uncertainty with respect to the final profits. For the variety of cases that can occur, we characterize the sets of equilibria, and provide procedures to determine them. The analysis proposed can also be applied to study situations in a deterministic setting with simultaneous multiple scenarios, and to the analysis of multiple criteria duopolistic competition.

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