Abstract
In resource contribution games, a class of non-cooperative games, the players want to obtain a bundle of resources and are endowed with bags of bundles of resources that they can make available into a common for all to enjoy. Available resources can then be used towards their private goals. A player is potentially satisfied with a profile of contributed resources when his bundle could be extracted from the contributed resources. Resource contention occurs when the players who are potentially satisfied, cannot actually all obtain their bundle. The player’s preferences are always single-minded (they consider a profile good or they do not) and parsimonious (between two profiles that are equally good, they prefer the profile where they contribute less). What makes a profile of contributed resources good for a player depends on their attitude towards resource contention. We study the problem of deciding whether an outcome is a pure Nash equilibrium for three kinds of players’ attitudes towards resource contention: public contention-aversity, private contention-aversity, and contention-tolerance. In particular, we demonstrate that in the general case when the players are contention-averse, then the problem is harder than when they are contention-tolerant. We then identify a natural class of games where, in presence of contention-averse preferences, it becomes tractable, and where there is always a Nash equilibrium.
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An implementation of the models and algorithms presented in this paper is available at https://bitbucket.org/troquard/irgpy/.
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The author wishes to thank the reviewers of this journal for pressing on crucial points.
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Open access funding provided by Gran Sasso Science Institute - GSSI within the CRUI-CARE Agreement. This research was partially supported by a grant from the Free University of Bozen-Bolzano, Computations in Resource Aware Systems (CompRAS).
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A Software implementation: examples, and proofs details
A Software implementation: examples, and proofs details
An implementation is available at https://bitbucket.org/troquard/irgpy/src/master/. Suffixes pct stand for (parsimonious) contention-tolerant preferences (\(\textsf{ct}\)), pca for (parsimonious public) contention-averse preferences (\(\textsf{ca}\)), and ppca for (parsimonious) private contention-averse preferences (\(\textsf{pca}\)).
1.1 A.1 Simple example
We begin with a simple series of examples. Based on the same RCG (more precisely, an RCGBAR), we examine the differences in the sets of Nash equilibria when changing the kind of preferences.
Example 1
Player 1, Ann, is endowed with \(\epsilon _1 = \{A,B\}\) and has the objective \(\gamma _1 = A\). Player 2 and player 3, both identical Bob, are endowed with \(\epsilon _2 = \epsilon _3 = \{A, B, B\}\) and have the objective \(\gamma _2 = \gamma _3 = B\).
The following script shows how the software can be used to determine the set of Nash equilibria in the game when considering (public) contention-averse preferences.
Example 11
(continued) Continuing Example 11, we can see the effect on the set of Nash equilibria of a change from contention-averse preferences to contention-tolerant preferences.
Example 11
Continuing Example 11, we can see the effect on the set of Nash equilibria of a change from (public) contention-averse preferences to private contention-averse preferences.
1.2 A.2 Back to Example 1
We now demonstrate how the motivating example Example 1 is implemented in the software.
Example 1
(continued) In the first part of the example, with you, Bruno, and Carmen the game is formally defined as: \(\tilde{G}_{1.1} = (\{y, b, c\},\gamma _y = \textsf{w}\varvec{\cdot }\textsf{w}, \gamma _b = \textsf{w}\varvec{\cdot }\textsf{r}\varvec{\cdot }\textsf{o}, \gamma _c = \textsf{w}\varvec{\cdot }\textsf{r}\varvec{\cdot }\textsf{o}, \epsilon _y = \{\textsf{w}\varvec{\cdot }\textsf{w}\varvec{\cdot }\textsf{w}\varvec{\cdot }\textsf{w}\}, \epsilon _b = \{\textsf{r}, \textsf{r}, \textsf{r}\}, \epsilon _c = \{\textsf{o}, \textsf{o}\})\)
In the second part of the example, when Edward joins the party, the game is formally defined as: \(\tilde{G}_{1.2} = (\{y, b, c, e\},\gamma _y = \textsf{w}\varvec{\cdot }\textsf{w}, \gamma _b = \textsf{w}\varvec{\cdot }\textsf{r}\varvec{\cdot }\textsf{o}, \gamma _c = \textsf{w}\varvec{\cdot }\textsf{r}\varvec{\cdot }\textsf{o}, \gamma _e = \{\textsf{w}\varvec{\cdot }\textsf{w}\varvec{\cdot }\textsf{w}\}, \epsilon _y = \{\textsf{w}\varvec{\cdot }\textsf{w}\varvec{\cdot }\textsf{w}\varvec{\cdot }\textsf{w}\}, \epsilon _b = \{\textsf{r}, \textsf{r}, \textsf{r}\}, \epsilon _c = \{\textsf{o}, \textsf{o}\}, \epsilon _e = \emptyset )\)
Continuing the previous execution, we simply add a player and define a new game with the four players.
1.3 A.3 Back to Example 2
We now demonstrate how our running example Example 2 is implemented in the software.
Example 2
(continued) The scenario was formalized into the RCG \(\tilde{G}_2 = (\{a,b\},\gamma _a = \textsf{3G}\varvec{\cdot }\textsf{3G}\varvec{\cdot }\textsf{4G}, \gamma _b = \textsf{3G}\varvec{\cdot }\textsf{3G}\varvec{\cdot }\textsf{4G}\varvec{\cdot }\textsf{4G}, \epsilon _a = \{\textsf{3G},\textsf{3G},\textsf{3G}\varvec{3G}\}, \epsilon _b = \{\textsf{3G},\textsf{4G},\textsf{4G}\varvec{\cdot }\textsf{4G}\})\).
1.4 A.4 Deviations explanations of Proposition 1
We can use the verbose mode of our implementation to repeat the proof of Proposition 1.
Example 2
We considered the RCG G, with two players 1 and 2, where \(\epsilon _1 = \epsilon _2 = \{\textsf{A}\varvec{\cdot }\textsf{B}\}\), \(\gamma _1 = A\), and \(\gamma _2 = \textsf{B}\varvec{\cdot }\textsf{B}\), and showed that \(\textsf{NE} ^\textsf{pref}(G) = \emptyset \) for every kind of preferences. This can be also verified programmatically as follows.
1.5 A.5 Deviations explanations of Proposition 6
We can use the verbose mode of our implementation to repeat the proof of Prop. 6.
Example 3
We consider the RCGBAR G where \(\epsilon _1 = \emptyset \), \(\epsilon _2 = \emptyset \), \(\epsilon _3 = \{B, B\}\), \(\epsilon _4 = \{A, A\}\), \(\gamma _1 = \textsf{A}\varvec{\cdot }\textsf{A}\varvec{\cdot }\textsf{B}\varvec{\cdot }\textsf{B}\), \(\gamma _2=\textsf{A}\varvec{\cdot }\textsf{B}\), \(\gamma _3 = B\), and \(\gamma _4 = A\). \(\textsf{NE} ^\textsf{pca}(G)\) is empty.
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Troquard, N. Existence and verification of Nash equilibria in non-cooperative contribution games with resource contention. Ann Math Artif Intell 92, 317–353 (2024). https://doi.org/10.1007/s10472-023-09905-7
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DOI: https://doi.org/10.1007/s10472-023-09905-7