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A Simulator for Hedonic Games
Luke Harold Miles University of Kentucky, Lexington, United States
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What’s a hedonic game? A set of players and, for each player, a ranking of possible groups to join
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A Simulator for Hedonic Games Luke Harold Miles University of - - PowerPoint PPT Presentation
A Simulator for Hedonic Games Luke Harold Miles University of - - PowerPoint PPT Presentation
A Simulator for Hedonic Games Luke Harold Miles University of Kentucky, Lexington, United States 1 Whats a hedonic game? A set of players and, for each player, a ranking of possible groups to join 2 Example of a Hedonic Game Favorite
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Austin: AG >A A >A AGC >A AC Dr Goldsmith: AG >G G >G AGC >G GC Cory: AKC >C KC >C AC >C C
Example of a Hedonic Game
Favorite group Loathed group Possible partitions: {AGC} {AG, C} {A, GC} {AC, G} {A, G, C}
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Basic Notation
G = (N, {≥i : i ∈ N}) is a hedonic game. N is the (finite) set of players. Each ≥i is a ranking of the coalitions containing i. A class of hedonic games is any (finite or infinite) set of hedonic games. π is a partition of N. π(i) is the coalition in π containing i.
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N = {Austin, Dr G, Cory} π = {{Austin, Dr G}, {Cory}}
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Core stability
A (nonempty) coalition C blocks a partition π iff every player i in C would be happier in C than in π(i). i.e., C blocks π iff ∀i∈C: C >i π(i). π is core stable iff no possible coalition C ⊆ N blocks π.
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Austin: AG >A A >A AGC >A AC Dr Goldsmith: AG >G G >G AGC >G GC Cory: AKC >C KC >C AC >C C
Core Stability with Austin, Dr G, and Cory
Favorite group Loathed group Possible partitions: {AGC} {AG, C} {A, GC} {AC, G} {A, G, C}
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Core stable?
- No. AG blocks
Yes!
- No. G blocks
- No. A blocks
- No. AG blocks
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Austin: AG >A AC >A A >A AGC Dr Goldsmith: GC >G AG >G G >G AGC Cory: AC >C GC >C C >C AGC
Core Stability with Austin, Dr G, and Cory (version 2)
Favorite group Loathed group Possible partitions: {AGC} {AG, C} {A, GC} {AC, G} {A, G, C}
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Core stable?
- No. AG blocks
- No. GC blocks
- No. AC blocks
- No. AG blocks
- No. AG blocks
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Two frequently-asked questions
Given a class of hedonic games...
- 1. Is there always a core-stable partition?
- 2. If not, how hard is it to decide?
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Friend-Oriented Hedonic Games
Each player labels every other player as either a friend or an enemy. Ranking: More friends is a lot better; fewer enemies is a little better.
- 1. Is there always a core-stable partition? Yes
(Dimitrov, Borm, Hendrickx, Sung. 2006.)
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Enemy-Oriented Hedonic Games
Each player labels every other player as either a friend or an enemy. Ranking: Fewer enemies is a lot better; more friends is a little better.
- 1. Is there always a core-stable partition? Yes
(Dimitrov, Borm, Hendrickx, Sung. 2006.)
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Fractional Hedonic Games
Each player scores every other player. (e.g. Cory ranks Austin 3.46) Ranking: Higher average score is better.
- 1. Is there always a core-stable partition? No
- 2. How hard is it to decide? Σ2
p -complete!
(Aziz, Brandl, Brandt, Harrenstein, Olsen, Peters. 2017.)
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Altruistic Hedonic Games
Each player labels every other player as either a friend or an enemy. Ranking: “I’ll pick the coalition in which my friends and I are both happy.”
- 1. Is there always a core-stable partition? No
- 2. How hard is it to decide? Varies
(Nguyen, Rey, Rey, Rothe, Schend. 2016.)
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The Simulator
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Screenshot
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Make adjacency list
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Make partition
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Choose Class of Hedonic Games
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Choose Stability Notion
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Compute Scores
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