Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.
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Transcript of Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.
![Page 1: Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d305503460f94a0977f/html5/thumbnails/1.jpg)
Taxation and Stability in Cooperative Games
Yair Zick
Maria Polukarov
Nick R. Jennings
AAMAS 2013
![Page 2: Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d305503460f94a0977f/html5/thumbnails/2.jpg)
Cooperative TU Games
Agents divide into coalitions; generate profit. 1
6
4
5
3
2Coalition members can freely divide profits.
How should profits
be divided?
$5
$3
$2
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TU Games - NotationsAgents: N = {1,…,n}Coalition: S µ NCharacteristic function: v: 2N → RA TU game G = hN,vi is
anonymous, if the value of a coalition is only a function of its size.
A TU game is monotone, if the value of a coalition can only increase by adding more agents to it.
![Page 4: Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d305503460f94a0977f/html5/thumbnails/4.jpg)
Payoffs
We assume that only the grand coalition N is formed.
Agents may freely distribute profits.An imputation is a vector x = (x1,
…,xn) such that Σi2N xi= v(N).
Individual rationality: each agent gets at least what she can make on her own: xi ≥ v({i})
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The CoreThe core is the set of all stable
outcomes: for all S µ N we have x(S) ¸ v(S); a game with a non-empty core is called stable.
May be empty in many games.Example: the 3-majority game.
Three players; any set of size two or more has a value of 1; singletons have a value of 0.
The total amount to be divided is 1; if (w.l.o.g.)
player 1 gets more than 0, then players 2 and 3 get a total payoff of less than 1.
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Stabilizing Games
The 3-Majority game has an empty core. However, if one reduces the value of all 2-
player coalitions by 1/3, the core becomes non-empty (giving 1/3 to each player is a stable outcome). Similarly: reduce the value of {2,3} to 0.
Reducing the value of some coalitions can result in a stable game.
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Our Work
We explore taxation methods (i.e. reductions in coalition value), that ensure stability.
What is the minimal amount of tax required in order to stabilize a game? Which taxation schemes are optimal? When are known taxation schemes
optimal?
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Some Background
Taxation is not new "-core: coalition values reduced by ". Least-core: corresponds to the "*-core,
where "* is the smallest " for which the "-core is not empty.
Reliability extensions: each agent i survives with probability ri; the value of a coalition is reduced to its expected value.
Myerson graphs, etc…
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1
2 3
x1 = v({1})
x2 = v(N
) − v({1,3})
x2 = v({2})
x1 = v(N) − v({2,3})
x3 = v({3})
x3 = v(N) − v({1,2})
x1 + x2 + x3 = v(N)
![Page 10: Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d305503460f94a0977f/html5/thumbnails/10.jpg)
1
2 3
x1 = v({1})
x2 = v(N
) − v({1,3})
x2 = v({2})
x1 = v(N) − v({2,3})
x3 = v({3})
x3 = v(N) − v({1,2})
x1 + x2 + x3 = v(N)
""
"
"
"
"
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1
2 3
x1 = v({1})
x2 = v(N
) − v({1,3})
x2 = v({2})
x1 = v(N) − v({2,3})
x3 = v({3})
x3 = v(N) − v({1,2})
x1 + x2 + x3 = v(N)
"
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Exploring Taxation
Methods
Given a game G = hN,vi, we say that G’ ≤ G if v(S) ≥ v’(S) for all S µ N.
A game G’ is maximal-stable w.r.t. G if G’ ≤ G It has a non-empty core If G’’ is stable and G’ ≤ G’’ ≤ G, then one
of the inequalities holds with equality.
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Exploring Taxation
Methods
Increasing the value of any coalition results in losing either stability or dominance.
Observations: Maximal-stable games still distribute v(N) to the
agents. They are defined by a single vector
x = (x1,…,xn); the value of each coalition S is
min{v(S),x(S)}.
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Optimal Taxation Schemes
The set of dominated games with a non-empty core is a convex polyhedron denoted S(G).
We are interested in the set of games that minimizes the total tax taken.
These games are said to have optimal taxation schemes.
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Optimal Taxation Schemes
We characterize the optimal taxation scheme for anonymous games.
Given an anonymous game, where v(S) = f(|S|), the optimal taxation scheme is given by reducing the value of each coalition to min(f(|S|),f(|N|)/|S|).
Good for small coalitions; bad for large coalitions.
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Existing Taxation Schemes
Suppose that a central authority wants to implement a taxation scheme.What conditions must hold in order for this taxation scheme to be optimal?
We find conditions on the underlying cooperative game which ensure this for the "-core and for reliability extensions.
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Conclusions & Future Work
Given a class of cooperative games, what taxation schemes would be optimal?
How much are we “over-taxing” by using a given taxation scheme?
Other ways of measuring total taxation.Computational complexity: efficient
computation of optimal taxes?
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Thank you!Questions?
P.S.: I am on the job market!